[style=none] (22) at (-2.5, -3) ;
[style=none] (31) at (-8, 0.75) ;
[style=none] (32) at (-6.25, 0.75) ;
[style=none] (33) at (-5.25, -0.25) ;
[style=none] (34) at (-5.25, -2) ;
[style=none] (35) at (-8, -0.75) ;
[style=none] (36) at (-6.75, -2) ;
[style=none] (37) at (-7.25, -0.75) ;
[style=none] (38) at (-6.75, -1.25) ;

(0) to (1);
(1) to (2);
(6) to (2);
(2) to (3);
(3) to (4);
(5) to (3);
[style=e3] (8.center)
to [bend left=90, looseness=1.75] (7.center)
to [bend left=90, looseness=1.75] cycle;
[style=e5] (10.center)
to [bend left=90, looseness=1.75] (9.center)
to [bend left=90, looseness=1.75] cycle;
[style=e6] (12.center)
to [bend left=90, looseness=1.75] (11.center)
to [bend left=90, looseness=1.75] cycle;
[style=e1] (14.center)
to [bend left=90, looseness=1.75] (13.center)
to [bend left=90, looseness=1.75] cycle;
[style=e3] (17.center)
to (15.center)
to [bend left=270, looseness=1.75] (16.center)
to (18.center)
to [bend right=90, looseness=1.75] cycle;
[style=e4] (21.center)
to (19.center)
to [bend left=270, looseness=1.75] (20.center)
to (22.center)
to [bend right=90, looseness=1.75] cycle;
[style=e2] (33.center)
to [bend right=45, looseness=1.25] (32.center)
to (31.center)
to [bend right=90, looseness=1.75] (35.center)
to (37.center)
to [bend left=45, looseness=1.50] (38.center)
to (36.center)
to [bend right=90, looseness=1.50] (34.center)
to cycle;

[scale=0.6,baseline = 0]
[style=label-s] (39) at (3, 3) $0$;
[style=label-s] (40) at (6, 2.5) $3'$;
[style=label-s] (41) at (6.5, 3) $5'$;
[style=label-s] (42) at (7, 3.5) $6'$;
[style=label-s] (43) at (9.75, 3) $0'$;
[style=label-s] (44) at (3, 0) $2$;
[style=label-s] (45) at (6, 0) $3'$;
[style=label-s] (46) at (7, 1) $4$;
[style=label-s] (47) at (9.75, 0.5) $0$;
[style=label-s] (48) at (6, -3) $0$;
[style=label-s] (49) at (3, -3) $1$;
[style=none] (50) at (8.75, 0.5) ;
[style=none] (51) at (8.75, 3) ;
[style=none] (52) at (4.5, 3) ;
[style=none] (53) at (6, -1.5) ;
[style=label-s] (55) at (2.25, 0.5) $1\oplus 1$;
[style=label-s] (56) at (7.75, 1.25) $1\oplus 1$;
[style=none] (57) at (4.25, 1.75) ;
[style=none] (59) at (8.25, 1.75) ;
[style=label-w] (60) at (4.75, 0) $e_2$;
[style=label-w] (61) at (4.75, 0.5) $e_1$;
[style=label-w] (62) at (4.25, 1.75) $d$;

[style=m1] (39) to (44);
[style=m1] (44) to (46);
[style=m1, in=180, out=0] (46) to (50.center);
[style=m1, in=0, out=-180] (50.center) to (45);
[style=m1] (50.center) to (47);
[style=m1] (51.center) to (43);
[style=m1, in=-180, out=-15, looseness=1.25] (42) to (51.center);
[style=m1] (51.center) to (41);
[style=m1, in=-180, out=0] (40) to (51.center);
[style=m1] (49) to (48);
(44) to (49);
[style=e4] (39) to (41);
[style=e4, in=180, out=0] (52.center) to (42);
[style=e4, in=180, out=0] (52.center) to (40);
[style=e2] (48) to (45);
[style=e2, in=-90, out=90] (53.center) to (46);
(44) to (45);
(43) to (47);
(40) to (45);
[style=e5] (41) to (46);
[style=e6] (42) to (46);
[style=d1] (44) to (48);
[style=d2, in=135, out=-45] (39) to (57.center);
[style=d2] (57.center) to (45);
[style=d2, in=150, out=-30, looseness=1.25] (57.center) to (46);
[style=d4] (41) to (59.center);
[style=d4] (59.center) to (47);
[style=d4, in=-45, out=150] (59.center) to (42);
[style=d4, in=150, out=-30] (40) to (59.center);

On the left is a graph with a maximal nested collection and on the right is a subset of edges of the snake graph $\mathcal{G}_0$ which violates Condition (<ref>) of Definition <ref>.  We suppress the edge weights and diagonal labels for clarity.

§.§ Statement of Main Theorem

Given a snake graph $\mathcal{G}$, let $\ell(\mathcal{G}):=\prod
_{d\in D} Y_{W(d)}$ where $D$ is the set of diagonals of $\mathcal{G}$.  Let $\chi(\mathcal{G}) := \frac{1}{\ell(G)} \sum_{P \in \mathcal{P}} \wt(P)$ where $\mathcal{P}$ is the set of admissible matchings of $\mathcal{G}$ and $\wt(P):=\prod_{e\in P}\wt(e)$ is the product of the weights of the edges in $P$.

For any cluster $\mathcal{I}$ and any weakly rooted set $S$ with respect to $\mathcal I$,
\[
\chi(\mathcal{G}_S) = Y_S.
\]

Below is the set of admissible matchings, with their weights, of the graph $\mathcal G_{345680}$ constructed in <Ref>. Recall in Example <ref> we demonstrated another matching of this graph which respected valence of all vertices but which was not admissible.

e1=[-, color=1, line width=1, opacity = 0.4]
e2=[-, color=2, line width=1, opacity = 0.4]
e3=[-, color=3, line width=1, opacity = 0.4]
e4=[-, color=4, line width=1, opacity = 0.4]
e5=[-, color=5, line width=1, opacity = 0.4]
e6=[-, color=6, line width=1, opacity = 0.4]
e7=[-, color=7, line width=1, opacity = 0.4]
e8=[-, color=8, line width=1, opacity = 0.4]
e9=[-, color=9, line width=1, opacity = 0.4]
e0=[-, color=0, line width=1, opacity = 0.4]
d1=[-, color=1, line width=1.2, loosely dashed, opacity = 0.4]
d2=[-, color=2, line width=1.2, loosely dashed, opacity = 0.4]
d3=[-, color=3, line width=1.2, loosely dashed, opacity = 0.4]
d4=[-, color=4, line width=1.2, loosely dashed, opacity = 0.4]
d5=[-, color=5, line width=1.2, loosely dashed, opacity = 0.4]
d6=[-, color=6, line width=1.2, loosely dashed, opacity = 0.4]
d7=[-, color=7, line width=1.2, loosely dashed, opacity = 0.4]
d8=[-, color=8, line width=1.2, loosely dashed, opacity = 0.4]
d9=[-, color=9, line width=1.2, loosely dashed, opacity = 0.4]
d0=[-, color=0, line width=1.2, loosely dashed, opacity = 0.4]

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) ${I_4}$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) ${I_1}$;
[style=label-s] (29) at (0.75, 3.75) ${I_2}$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4.25) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) ${I_7}$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) ${Y_0^2}$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $Y_{I_2}$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $Y_{I_7}$;
[style=none] (58) at (-4, -5) $Y_{123456780}Y_2Y_{0}^2$;

[style=m1] (0) to (1);
[opacity=0.4] (1) to (3);
[style=e8] (0) to (2);
[opacity=0.4] (7) to (8);
[style=m1] (8) to (9);
[style=e1] (9) to (6);
[opacity=0.4] (6) to (7);
[style=e1] (11) to (6);
[style=m1] (2) to (16.center);
[style=m1, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=m1, in=-180, out=0] (16.center) to (4);
[style=m1, in=120, out=0] (16.center) to (5);
[style=m1, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=e4, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=e4] (15.center) to (5);
[style=e7] (14.center) to (4);
[style=e7] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=e4, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=e7] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=e4, in=90, out=-60] (38) to (45.center);
[style=e4, in=120, out=-90] (45.center) to (44);
[style=e4] (39) to (45.center);
[style=e0, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=e0] (46.center) to (41);
[style=m1] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=e4] (45.center) to (0);
[style=e0] (46.center) to (1);
[style=d7] (47) to (1);
[style=m1] (44) to (1);
[style=m1] (39) to (41);
[style=m1] (0) to (7);
[style=e4] (15.center) to (11);

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) $I_4$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) $I_1$;
[style=label-s] (29) at (0.75, 3.75) $I_2$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4.25) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) $I_7$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) $Y_0^2$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $Y_{I_2}$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $Y_{I_7}$;
[style=none] (58) at (-4, -5) ;
[style=none] (59) at (-4, -5) $Y_8 Y_{45670} Y_0^2$;

[style=m1] (0) to (1);
[opacity=0.4] (1) to (3);
[style=m1] (0) to (2);
[opacity=0.4] (7) to (8);
[style=m1] (8) to (9);
[style=e1] (9) to (6);
[style=m1] (6) to (7);
[style=e1] (11) to (6);
[style=e3] (2) to (16.center);
[style=e3, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=e3, in=-180, out=0] (16.center) to (4);
[style=e3, in=120, out=0] (16.center) to (5);
[style=e3, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=m1] (15.center) to (5);
[style=e7] (14.center) to (4);
[style=e7] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=e7] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=e4, in=90, out=-60] (38) to (45.center);
[style=e4, in=120, out=-90] (45.center) to (44);
[style=e4] (39) to (45.center);
[style=e0, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=e0] (46.center) to (41);
[style=m1] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=e4] (45.center) to (0);
[style=e0] (46.center) to (1);
[style=d7] (47) to (1);
[style=m1] (44) to (1);
[style=m1] (39) to (41);
[opacity=0.4] (0) to (7);
[style=m1] (15.center) to (11);

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) $I_4$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) $I_1$;
[style=label-s] (29) at (0.75, 3.75) $I_2$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4.25) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) $I_7$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) $Y_0^2$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $Y_{I_2}$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $I_7$;
[style=none] (58) at (-3.75, -5) ;
[style=none] (59) at (-3.75, -5) $Y_8Y_{45670}^2Y_{560}Y_{12}$;

[opacity=0.4] (0) to (1);
[opacity=0.4] (1) to (3);
[style=m1] (0) to (2);
[style=m1] (7) to (8);
[opacity=0.4] (8) to (9);
[style=m1] (9) to (6);
[opacity=0.4] (6) to (7);
[style=e1] (11) to (6);
[style=e3] (2) to (16.center);
[style=e3, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=e3, in=-180, out=0] (16.center) to (4);
[style=e3, in=120, out=0] (16.center) to (5);
[style=e3, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=m1] (15.center) to (5);
[style=e7] (14.center) to (4);
[style=e7] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=e7] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=m1, in=90, out=-60] (38) to (45.center);
[style=m1, in=120, out=-90] (45.center) to (44);
[style=m1] (39) to (45.center);
[style=m1, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=m1] (46.center) to (41);
[opacity=0.4] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=m1] (45.center) to (0);
[style=m1] (46.center) to (1);
[style=d7] (47) to (1);
[opacity=0.4] (44) to (1);
[opacity=0.4] (39) to (41);
[opacity=0.4] (0) to (7);
[style=m1] (15.center) to (11);

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) $I_4$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) $I_1$;
[style=label-s] (29) at (0.75, 3.75) $I_2$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4.25) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) $I_7$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) $Y_0^2$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $Y_{I_2}$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $I_7$;
[style=none] (58) at (-3.75, -5) ;
[style=none] (59) at (-3.75, -5) $Y_8Y_{45670}Y_0^2Y_{12}$;

[style=m1] (0) to (1);
[opacity=0.4] (1) to (3);
[style=m1] (0) to (2);
[style=m1] (7) to (8);
[opacity=0.4] (8) to (9);
[style=m1] (9) to (6);
[opacity=0.4] (6) to (7);
[style=e1] (11) to (6);
[style=e3] (2) to (16.center);
[style=e3, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=e3, in=-180, out=0] (16.center) to (4);
[style=e3, in=120, out=0] (16.center) to (5);
[style=e3, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=m1] (15.center) to (5);
[style=e7] (14.center) to (4);
[style=e7] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=e7] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=e4, in=90, out=-60] (38) to (45.center);
[style=e4, in=120, out=-90] (45.center) to (44);
[style=e4] (39) to (45.center);
[style=e0, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=e0] (46.center) to (41);
[style=m1] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=e4] (45.center) to (0);
[style=e0] (46.center) to (1);
[style=d7] (47) to (1);
[style=m1] (44) to (1);
[style=m1] (39) to (41);
[opacity=0.4] (0) to (7);
[style=m1] (15.center) to (11);

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) $I_4$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) ${I_1}$;
[style=label-s] (29) at (0.75, 3.75) $I_2$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4.25) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) $I_7$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) $Y_0^2$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $Y_{I_2}$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $Y_{I_7}$;
[style=none] (58) at (-3.75, -5) $Y_2Y_{5670}Y_8Y_0^2Y_{12}$;

[opacity=0.4] (0) to (1);
[opacity=0.4] (1) to (3);
[style=m1] (0) to (2);
[opacity=0.4] (7) to (8);
[style=m1] (8) to (9);
[style=e1] (9) to (6);
[opacity=0.4] (6) to (7);
[style=m1] (11) to (6);
[style=e3] (2) to (16.center);
[style=e3, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=e3, in=-180, out=0] (16.center) to (4);
[style=e3, in=120, out=0] (16.center) to (5);
[style=e3, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=e4, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=e4] (15.center) to (5);
[style=m1] (14.center) to (4);
[style=m1] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=e4, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=m1] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=e4, in=90, out=-60] (38) to (45.center);
[style=e4, in=120, out=-90] (45.center) to (44);
[style=e4] (39) to (45.center);
[style=e0, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=e0] (46.center) to (41);
[style=m1] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=e4] (45.center) to (0);
[style=e0] (46.center) to (1);
[style=d7] (47) to (1);
[style=m1] (44) to (1);
[style=m1] (39) to (41);
[style=m1] (0) to (7);
[style=e4] (15.center) to (11);

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) $I_4$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) $I_1$;
[style=label-s] (29) at (0.75, 3.75) $I_2$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) $I_7$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) $Y_0^2$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $I_2$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $Y_{I_7}$;
[style=none] (58) at (-3.75, -5) ;
[style=none] (59) at (-3.75, -5) $Y_{123456780}Y_{2}Y_{45670}Y_{560}$;

[opacity=0.4] (0) to (1);
[opacity=0.4] (1) to (3);
[style=e8] (0) to (2);
[opacity=0.4] (7) to (8);
[style=m1] (8) to (9);
[style=e1] (9) to (6);
[opacity=0.4] (6) to (7);
[style=e1] (11) to (6);
[style=m1] (2) to (16.center);
[style=m1, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=m1, in=-180, out=0] (16.center) to (4);
[style=m1, in=120, out=0] (16.center) to (5);
[style=m1, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=e4, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=e4] (15.center) to (5);
[style=e7] (14.center) to (4);
[style=e7] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=e4, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=e7] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=m1, in=90, out=-60] (38) to (45.center);
[style=m1, in=120, out=-90] (45.center) to (44);
[style=m1] (39) to (45.center);
[style=m1, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=m1] (46.center) to (41);
[opacity=0.4] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=m1] (45.center) to (0);
[style=m1] (46.center) to (1);
[style=d7] (47) to (1);
[opacity=0.4] (44) to (1);
[opacity=0.4] (39) to (41);
[style=m1] (0) to (7);
[style=e4] (15.center) to (11);

[style=label] (0) at (-7, 4.5) $3$;
[style=label] (1) at (-2.25, 4.5) $4$;
[style=label] (2) at (-7, 0.5) $8'$;
[style=label] (3) at (-2.25, 1.25) $9$;
[style=label] (4) at (-3, 0.5) $7'$;
[style=label] (5) at (-3.75, -0.25) $0'$;
[style=label] (6) at (-1.5, 2) $1'$;
[style=label] (7) at (-1.5, 5.5) $1$;
[style=label] (8) at (2, 5.5) $2$;
[style=label] (9) at (2, 2) $3$;
[style=label] (11) at (-3.75, -3.5) $3$;
[style=none] (12) at (0, 4) ;
[style=label-s] (13) at (-5, 2.5) $I_4$;
[style=none] (14) at (-3, 2) ;
[style=none] (15) at (-3.75, -1.75) ;
[style=none] (16) at (-5.75, 0.5) ;
[style=label-s] (22) at (-2.25, -1) $Y_{I_1}$;
[style=label-s] (23) at (0.25, 1.5) $Y_{I_1}$;
[style=label-s] (25) at (-7.5, 2.25) $Y_{I_8}$;
[style=label-s] (27) at (-3.3, 4) $1\oplus 2$;
[style=label-s] (28) at (-4.5, 3.75) $I_1$;
[style=label-s] (29) at (0.75, 3.75) $I_2$;
[style=label-s] (30) at (-3.75, -4) $0\oplus 1$;
[style=label-s] (31) at (-8, 4.25) $2\oplus 0$;
[style=label] (38) at (-7.5, 8.5) $7'$;
[style=label] (39) at (-7, 9) $0'$;
[style=label] (40) at (-2.75, 8.5) $7$;
[style=label] (41) at (-2.25, 9) $0'$;
[style=label] (44) at (-6.5, 5) $9$;
[style=none] (45) at (-7, 6.75) ;
[style=none] (46) at (-2.25, 6.5) ;
[style=label-s] (47) at (-4.75, 6.75) $I_7$;
[style=label-s] (51) at (-1.75, 6.75) $Y_{I_0}$;
[style=label-s] (52) at (-4.75, 9.5) $Y_0^2$;
[style=label-s] (53) at (-5, 0) $Y_{I_3}$;
[style=label-s] (54) at (-4.25, 5.5) $Y_{I_2}$;
[style=label-s] (55) at (-4.25, -1.75) $Y_{I_4}$;
[style=label-s] (56) at (-7.5, 6.75) $Y_{I_4}$;
[style=label-s] (57) at (-3.5, 2.25) $Y_{I_7}$;
[style=none] (58) at (-3.75, -5) $Y_8Y_{45670}^2Y_{560}$;

[opacity=0.4] (0) to (1);
[opacity=0.4] (1) to (3);
[style=m1] (0) to (2);
[opacity=0.4] (7) to (8);
[style=m1] (8) to (9);
[style=e1] (9) to (6);
[style=m1] (6) to (7);
[style=e1] (11) to (6);
[style=e3] (2) to (16.center);
[style=e3, in=-150, out=0, looseness=0.75] (16.center) to (3);
[style=e3, in=-180, out=0] (16.center) to (4);
[style=e3, in=120, out=0] (16.center) to (5);
[style=e3, in=210, out=0, looseness=0.75] (16.center) to (6);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (4);
[style=m1] (15.center) to (5);
[style=e7] (14.center) to (4);
[style=e7] (14.center) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=m1, in=-120, out=90, looseness=0.50] (15.center) to (3);
[style=d1] (0) to (6);
[style=e7] (14.center) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=m1, in=90, out=-60] (38) to (45.center);
[style=m1, in=120, out=-90] (45.center) to (44);
[style=m1] (39) to (45.center);
[style=m1, in=90, out=-60, looseness=0.75] (40) to (46.center);
[style=m1] (46.center) to (41);
[opacity=0.4] (38) to (40);
[style=d7, in=135, out=-15, looseness=0.75] (38) to (47);
[style=d7] (47) to (39);
[style=m1] (45.center) to (0);
[style=m1] (46.center) to (1);
[style=d7] (47) to (1);
[opacity=0.4] (44) to (1);
[opacity=0.4] (39) to (41);
[opacity=0.4] (0) to (7);
[style=m1] (15.center) to (11);

e1=[-, color=1, line width=1]
e2=[-, color=2, line width=1]
e3=[-, color=3, line width=1]
e4=[-, color=4, line width=1]
e5=[-, color=5, line width=1]
e6=[-, color=6, line width=1]
e7=[-, color=7, line width=1]
e8=[-, color=8, line width=1]
e9=[-, color=9, line width=1]
e0=[-, color=0, line width=1]
d1=[-, color=1, line width=1.2, loosely dashed]
d2=[-, color=2, line width=1.2, loosely dashed]
d3=[-, color=3, line width=1.2, loosely dashed]
d4=[-, color=4, line width=1.2, loosely dashed]
d5=[-, color=5, line width=1.2, loosely dashed]
d6=[-, color=6, line width=1.2, loosely dashed]
d7=[-, color=7, line width=1.2, loosely dashed]
d8=[-, color=8, line width=1.2, loosely dashed]
d9=[-, color=9, line width=1.2, loosely dashed]
d0=[-, color=0, line width=1.2, loosely dashed]
none=[inner sep=0mm]

We have $\ell(\mathcal G_{345680})=Y_{2}Y_{12}Y_{5670}Y_{45670}$, so <Ref> tells us that
\begin{align*}
Y_{345680}=&\ \frac{1}{Y_{2}Y_{12}Y_{5670}Y_{45670}}\left(Y_{123456780}Y_2Y_{0}^2+Y_8Y_{45670}Y_{0}^2+Y_8Y_{45670}^2Y_{560}Y_{12}+Y_8Y_{45670}Y_{0}^2Y_{12}+\right.\\
&\ \left.Y_2Y_{5670}Y_8Y_{0}^2Y_{12}+Y_{123456780}Y_2Y_{45670}Y_{560}+Y_8Y_{45670}^2Y_{560}\right)
\end{align*}

We can check this expression by starting with the cluster given by the maximal nested collection in <Ref> and mutating at $Y_{5670}$, the $Y_{12}$, then $Y_{2}$, then $Y_{45670}$ with the mutation equations given in Lemma 4.11 of <cit.>.

* Note that <Ref> does not guarantee an expression for $Y_S$ in terms of the cluster variables $\{Y_I\ |\ I\in\mathcal I\}$.  In particular, the expression in the previous example contains $Y_0$ and $\{0\}$ is not a set in the original nested collection in <Ref>.

* The denominator of the expansion we obtain from <Ref> is guaranteed to be square-free by Proposition <ref>.  However, if we expand any  variables that  are not in the original  cluster, we no longer necessarily have a square-free denominator.  For example, if we expand $Y_0$ in terms of the cluster in <Ref>, the denominator of the expression is no longer square-free.

We defer the proof of Theorem <ref> to Sections <ref> and <ref>, as it is quite technical.

§ POSITIVITY

As noted above, Theorem <ref> does not necessarily given an expansion for cluster variable $Y_S$ in terms of the nested collection $\mathcal I$ because the edges in the snake graph may have weights that are not expressed in terms of $\mathcal I$.  The following theorem characterizes when our formula gives an expansion purely in terms of the cluster from the maximal nested collection.

If $S$ is a weakly rooted set with respect to $\mathcal I$, then the snake graph $\mathcal{G}_S$ has edges weighted only by products of variables in $\{Y_I\ |\ I\in\mathcal I\}$ if and only if for every vertex $v\in \overline{S}$, the connected components of the set $I_{j+1}^i(v)\setminus[v,a_j^i(v)]$ are elements of $\mathcal I$ for all $1\leq i\leq r$ and $1\leq j < c_i(v)$ where $I_j^i(v)\nsubseteq S$.

First suppose that for every vertex $v\in \overline{S}$, the set $I_{j+1}^i(v)\setminus[v,a_j^i(v)]$ consists of connected components which are elements of $\mathcal I$ for all $1\leq j < c_i(v)$ where $I_j^i(v)\nsubseteq S$. The snake graph $\mathcal{G}_{\overline{S}}$ is a gluing of $\mathcal{G}_{v}^S$ for all $v\in\overline {S}$.  From the list of edge weights in <Ref>, the only edges in $\mathcal{G}_{v}$ with weights that are not 1 or in the set $\{Y_I\ |\ I\in\mathcal I\}$ have weight $Y_C^2$ where $C$ is a connected component of $I_{j+1}^i(v)\setminus[v,a_j^i(v)]$.  Thus, the only edges in $\mathcal{G}_{\overline{S}}$ with weights that are not 1 or in the set $\{Y_I\ |\ I\in\mathcal I\}$ have weight $Y_C^2$ where $C$ is a connected component of $I_{j+1}^i(v)\setminus[v,a_j^i(v)]$ for some $v\in\overline{S}$.  Such an edge is adjacent to diagonals $I_j^i(v)$ and $I_{j+1}^i(v)$ in $\mathcal{G}_v$ and thus is adjacent to diagonals $I_j^i(v)$ and $I_{j+1}^i(v)$ in $\mathcal{G}_{\overline{S}}$.  If $I_j^i(v)\subseteq S$, then this edge gets deleted in creating $\mathcal{G}_v^S$ ,which is a subgraph of $\mathcal{G}_{\overline{S}}$.  Otherwise, the weight of the edge is a product of variables in $\{Y_I\ |\ I\in\mathcal I\}$ by our assumption.

For the other direction, suppose there is a vertex $v\in \overline{S}$ such that the set $I_{j+1}^i(v)\setminus[v,a_j^i(v)]$ has a connected component $C\not\in\mathcal I$ for some $1\leq j < c_i(v)$ where $I_j^i(v)\nsubseteq S$.  Let $u$ be a leaf in $\Gamma'$ such that $[u,v]$ has nonempty intersection with $C$.  Consider some pair $\ell,k$ such that $u=\ell$ if $i=1$ and  $u=k$ if $i>1$.  By definition, $I^i_{j+1}(v)$ is a connected component of $I^i_j(v)\setminus\{a_j^i(v)\}$.  Thus the neighbor of $I^i_{j+1}(v)$ that is not $v$ in $[\ell,k]$ and the neighbor of $I^i_j(v)$ that is not $v$ in $[\ell,k]$ are the same.  In other words, $b_{i,j}^{\ell,k}=b_{i,j+1}^{\ell,k}$.  The vertex $b_{i,j+1}^{\ell,k}$ is not adjacent to a vertex on the path $[v,a_j^i(v)]$ as this would force $[u,v]\cap C$ to be empty.  Thus, $H_{\ell,k}$ has an edge weighted $Y_C^2$ adjacent to diagonals $I^i_j(v)$ and $I^i_{j+1}(v)$.  This means $\mathcal{G}_v$ has an edge weighted $Y_C^2$ adjacent to diagonals $I^i_j(v)$ and $I^i_{j+1}(v)$ and therefore $\mathcal{G}_{\overline{S}}$ has an edge weighted $Y_C^2$ adjacent to diagonals $I^i_j(v)$ and $I^i_{j+1}(v)$.  Since $I_j^i(v)\nsubseteq S$, this edge is not removed when creating $\mathcal{G}_v^S$, so it remains in $\mathcal{G}_{S}$. We can conclude that there is an edge weighted $Y_C^2$ in $\mathcal{G}_S$.  Our proofs in Sections <ref>, <ref>, <ref>, and <ref> show that this edge gets used in an admissible matching. Since $Y_C^2$ is not a product of variables in $\{Y_I\ |\ I\in\mathcal I\}$, the statement holds.

As mentioned in the previous section, the expansion for $Y_{34}$ in <Ref> contains cluster variables that are not in the original cluster $\{Y_I\ |\ \in\mathcal{I}\}$.  Notice that vertex 4 is in $\overline{S}$ and $I^1_2(4)\setminus[4,a^1_1(4)]=\{5,6,0\}\setminus[4,7]=\{0\}$.  Since $\{0\}$ is not in $\mathcal{I}$, the fact that the snake graph does not give an expansion in terms of the variables $\{Y_I\ |\ \in\mathcal{I}\}$ is consistent with <Ref>.

Together, Theorem <ref> and Theorem <ref> produce the following corollary:

If  $S$ is a weakly rooted set with respect to $\mathcal I$ and for every vertex $v\in \overline{S}$, the set $I_{j+1}^i(v)\setminus[v,a_j^i(v)]$ consists of connected components which are elements of $\mathcal I$ for all $1\leq i\leq r$ and $1\leq j < c_i(v)$ where $I_j^i(v)\nsubseteq S$, then $Y_S$ can be written as a Laurent polynomial with positive coefficients in the cluster variables $\{Y_I\ |\ I\in\mathcal I\}$.

§ HYPER $T$-PATHS

In <cit.>, we constructed hyper $T$-paths for any set $S$ with respect to a rooted cluster and gave an expansion formula for the variable $Y_S$ that was indexed by the set of all complete hyper $T$-paths for $S$. This setting was more restrictive than our current setting of weakly rooted sets with respect to an arbitrary cluster. One of the main distinctions is that, in terms of present notation, in <cit.> all singleton sets $\{v\}$ had $c_i(v) = 1$ for all $1 \leq i \leq r$.

In the setting of ordinary cluster algebras, there is a correspondence between $T$-paths and perfect matchings of snake graphs, described in <cit.>, where the vertices of the snake graph appear as nodes in the $T$-path and the edges used in the perfect matching and diagonals appear as connections between the nodes. For ordinary cluster algebras, $T$-paths are genuine path graphs. The spirit of this correspondence extends to the graph LP algebra setting, where we can now use our snake graph construction to describe a new, more general, version of hyper $T$-paths via the analogous correspondence with admissible matchings of $\mathcal{G}_S$, for any weakly rooted set $S$. Now, vertices of $\mathcal{G}_S$ continue to correspond to nodes, but the edges of an admissible matching and diagonals correspond to hyperedges and the resulting hyper $T$-path is a hypergraph.

To describe the resulting hyper $T$-paths, we modify Definition 4.3 of <cit.> to allow for our more general setting. Given a weakly rooted vertex subset $S$ of a tree $\Gamma$ with snake graph $\mathcal{G}_S$, let $\mathcal{B}$ denote the set of vertices of $\mathcal{G}_S$ with valence $0 \oplus b_k$. Note that vertices in $\mathcal{B}$ necessarily have $b_k > 0$ since vertices with valence $0 \oplus 0$ are deleted during the construction of $\mathcal{G}_S$. For any vertex subset $S$, recall that $N_{\Gamma'}(S)$ denotes the neighbors of $S$ in $\Gamma'$.

To state our modified definition, we also need some relevant terminology for hypergraphs. In an undirected hypergraph with vertex set $X$ and hyperedge set $E$, each $e \in E$ is a subset of $X$. Elements of $e$ are referred to as endpoints of $e$. A path on a hypergraph is a sequence $(v_0, e_1, v_1, \dots, v_{m-1}, e_m, v_{m})$ of distinct vertices and hyperedges such that $\{v_{i-1}, v_i \} \subseteq e_i$. Paths on a hypergraph correspond to paths, defined in the usual way, on its incidence graph. A cycle on a hypergraph is a path where $v_{0} = v_{m}$.

Let $S$ be a weakly rooted vertex subset of $\Gamma$ with snake graph $\mathcal{G}_S$, constructed as in Section <ref>. A complete hyper $T$-path for $S$ with respect to $\mathcal{I}$ is a set of nodes, which correspond to the vertices of $\mathcal{G}_S$, that are joined by connections, which are hyperedges that correspond to edges and diagonals of $\mathcal{G}_S$. Connections that correspond to edges of $\mathcal{G}_S$ are called odd and connections that correspond to diagonals of $\mathcal{G}_S$ are called even. Each odd connection has an associated weight that is the same as the weight of the corresponding edge in $\mathcal{G}_S$.  Each even connection has an associated label that is the same as the label of the corresponding diagonal in $\mathcal{G}_S$.  For an odd connection $c$ corresponding to edge $e$, we define $W(c)=W(e)$ and for an even connection $c$ corresponding to diagonal $d$, we define $W(c)=W(d)$. Each node has the same associated valence $a_k \oplus b_k$ as the vertex it corresponds to in $\mathcal{G}_S$. This collection of nodes and connections forms a connected diagram, such that:

* If a connection is weighted by $Y_I$, it joins nodes labeled by all of the vertices in $I$ with multiplicity one. If a connection is weighted by $Y_{I}^2$, then it joins a collection of nodes labeled by all vertices of $N_{\Gamma'}(I)$ except for the neighbor that lies on the path between an  element $v\in I$ and an  element of $u\in S$, with multiplicity two.  Note that this vertex does not depend on the choice of $v$ and $u$.

* There is a distinguished set of boundary nodes labeled by elements of $N_{\Gamma'}(S)\cup\mathcal{B}$?. The set of boundary nodes must contain exactly one node labeled by each vertex in $N_{\Gamma'}(S)$ and at most one node labeled by each vertex in $\mathcal{B}$. All other nodes are called internal nodes.

* A boundary node labeled by $v_k$ is adjacent to between $a_k$ and $a_k+b_k$ odd connections and no even connections.

*  An internal node labeled by $v_k$ is adjacent to exactly $a_k$ even connections and between $a_k$ and $a_k + b_k$ odd connections.

* If $x,y  \in N_{\Gamma'}(S)$ are such that there exist $v,w\in S$ with $x<_{\mathcal{I}} v$ and $y <_{\mathcal{I}} w$ (where possibly $v = w$), any path in any complete hyper $T$-path from boundary node $x$ to boundary node $y$ uses even connections labeled, in order, by $I_{x}$, $I_{u_\ell}$, $I_{u_{\ell-1}}, \dots, I_{u_1}, I_{w_1}, I_{w_2}, \dots, I_{w_k}, I_{y}$ where the path from $x$ to $y$ in $\Gamma'$ is $x, u_{\ell}, u_{\ell-1}, \dots, u_{1}, x \vee y, w_1, w_2, \dots, w_{k}, y$ for $k,\ell \geq 0$.

* If $x,y \in N_\Gamma(S)$ such that there exists $v \in S$ with $x <_{\mathcal{I}} v$ and $y >_{\mathcal{I}} m_{\mathcal{I}}(S)$, any path in any complete hyper $T$-path from boundary node $x$ to boundary node $y$ uses even connections that are labeled, in order, by $I_{x}, I_{u_\ell}, \dots, I_{u_2}$, where the path from $x$ to $y$ in $\Gamma'$ is $x, u_{\ell}, u_{\ell-1}, \dots, u_{1}, y$ for $\ell \geq 1$. If $\ell = 1$, then a path from $x$ to $y$ uses only the even connection $I_{x}$.

* If $x,y$ are boundary nodes and the path from $x$ to $y$ in $\Gamma'$ is $x, u_{\ell}, \dots, u_{k}, \dots, u_1,$ ${x\vee y},$ $ w_{1}, \dots, w_{g}, \dots, w_{h}, y$ where $u_{\ell}, \dots, u_{k-1} \in I \in \mathcal{I}$ such that $I \subseteq S$ and $w_{g+1}, \dots, w_{h} \in J \in \mathcal{I}$ such that $J \subseteq S$, then any path in the complete hyper $T$-path from $x$ to $y$ uses nodes labeled by the elements of $N_{\Gamma}(I_{j}^{i}(u_{k}))$ if $x \in I_{j}^{i}(u_{k})$, elements of $N_{\Gamma}(I_{j}^{i}(u_{k-1}))$ if $u_{k} \in I_{j}^{i}(u_{k-1})$, $\dots$, and elements of $N_{\Gamma}(I_{j}^{i}(w_{g}))$ if $y \in I_{j}^{i}(w_{g})$.

* There does not exist a cycle using connections $o_1, e_2,$ and $o_3$ such that $o_1, o_3$ are odd, $e_2$ is even, $o_1 \cap o_3 \neq \emptyset$, and $W(o_1) \subsetneq W(e_2) \subsetneq W(o_3)$.

*  There does not exist a cycle using connections $o_1,e_1,o_2,$ and $e_2$ such that $o_1, o_2$ are odd, $e_1, e_2$ are even, $W(o_1) = W(e_1)$, and $W(o_2) = W(e_2)$.

* If $\mathcal{G}_S$ contains a pair of boundary edges $e_1, e_2$ that are labeled with non-empty sets such that both share vertices with a diagonal $d$ but do not share any vertices with each other, and have $W(e_1) \subsetneq W(d) \subsetneq W(e_2)$, then there must be either no odd connections corresponding to $e_1$ and $e_2$ or there must be odd connections that correspond to both.

*  Suppose that $\mathcal{G}_{S}$ contains a pair of edges $e_1, e_2$ that share a vertex and that there exist other vertices $v_1,v_2$ in $\mathcal{G}_{S}$ such that $v_1$ is an endpoint of $e_1$ but not $e_2$ and $v_2$ is an endpoint of $e_2$ but not $e_1$.  If $v_1$ and $v_2$ are both endpoints of a diagonal labeled by $d$ where $W(e_1),W(e_2) \subseteq W(d)$ and $e_1,e_2$ are either both internal edges or both boundary edges incident to internal edges, then either both $e_1$ and $e_2$ have corresponding odd connections or neither have corresponding odd connections.

Let $\alpha$ denote a complete hyper $T$-path. We continue to define its weight as

\[ \wt(\alpha) := \left( \prod_{\textrm{odd connections  } c} \wt(c) \right) \left( \prod_{\textrm{even connections } c} Y_{W(c)} \right)^{-1}.  \]

Using this more general definition of a complete hyper $T$-path, our previous expansion formula from <cit.>  now extends to produce expansion formulas for all weakly rooted sets:

\[ Y_S = \sum_{\substack{\textrm{complete hyper} \\ T\textrm{-paths }\alpha \textrm{ for } S}} \wt(\alpha).\]

In our following examples, odd connections will be drawn with solid lines and even connections will be drawn with dashed lines.

Let us return to Example <ref> and consider the inadmissible matching shown in Figure <ref>. This inadmissible matching would correspond to the hyper $T$-path:

(1) at (0,0) $1$;
(2) at (2,0) $2$;
(3) at (4,-0.5) $1'$;
(4) at (4,0.5) $5$;
(5) at (6,-0.5) $3'$;
(6) at (8,-0.5) $2$;
(7) at (10,-0.5) $3$;
(8) at (6,1) $1'$;
(9) at (6,2) $3'$;
(10) at (8,2) 4';
(11) at (8,1) 6';
(12) at (10,1.5) $5$;
(13) at (12,2) 4;
(14) at (12,1) $6$;

(1) to (2);
[color=black!10!white,thick, line width = 10pt] (2) to (3);
[thick, densely dashed, color=blue] (2) to (3);
at (3,-0.6) ${I_1}$;
[color=black!10!white,thick, line width = 10pt] (2) to (4);
(2) to (4);
[color=black!10!white,thick, line width = 10pt] (3) to (5);
[color=black!10!white,thick, line width = 10pt] (4) to (5,-0.5);
[thick,red] (3) to (5);
[thick,red] (4) to (5,-0.5);
at (5.25,-0.2) $Y_{I_2}$;
[thick, densely dashed, color = 7] (5) to (6);
at (7,-0.8) ${I_3}$;
(6) to (7);
[thick,red, densely dashed] (4) to (8);
[thick,red, densely dashed] (5,0.75) to (9);
at (4.75,1) ${I_2}$;
(12) to (13);
(12) to (14);
[thick,densely dashed, purple] (10) to (12);
at (9,2) ${I_4}$;
[thick, densely dashed, green] (11) to (12);
at (9,1) ${I_6}$;
[thick,orange] (6.75,1.5) to (7.25,1.5);
[thick,orange] (9) to (6.75,1.5);
[thick,orange] (8) to (6.75,1.5);
[thick,orange] (7.25,1.5) to (10);
[thick,orange] (7.25,1.5) to (11);
at (7,1.75) $Y_{I_5}$;

Notice that this hyper $T$-path contains a highlighted sequence of connections $(o_1,e_2,o_3)$ where $o_1$ is the $2 - 5$ connection, $e_2$ is the $2 - 1'$ connection, and $o_3$ is the odd connection between $5, 1',$ and $3'$. Notice that $o_1$ and $o_3$ share an endpoint and that $W(o_1) = \emptyset$, $W(e_2) = I_1$, and $W(o_3) = I_2$. Because $W(o_1) \subsetneq W(e_2) \subsetneq W(o_3)$, this hyper $T$-path violates Condition (<ref>).

Next, let us consider the graph and maximal nested collection from Example <ref> and the snake graph $\mathcal{G}_{25}$ shown in Figure <ref>. This figure displays a non-admissible matching that corresponds to the following (invalid) hyper $T$-path:

(1) at (2,-1.5) $1$;
(2) at (2,0) $2$;
(3) at (4,0) $1'$;
(4) at (6,0.5) $3$;
(5) at (6,-0.5) $5$;
(6) at (8,0) $1'$;
(7) at (8,-1) $3$;

(8) at (10,0) $6'$;
(9) at (12,0) $5$;
(10) at (14,0.5) $1'$;
(11) at (14,-0.5) $3'$;
(12) at (16,0) $5$;
(13) at (18,-0.5) $6$;
(14) at (18,0.5) $4'$;
(15) at (18,2) $4$;

(1) to (2);
[thick, densely dashed, color=1] (2) to (3);
at (3,-0.4) ${I_1}$;
[thick, color=1,out = 45, in = 110] (2) to node[midway,above]$Y_{I_1}$ (6);
[thick, color=2] (3) to (5.25,0);
[thick, color=2] (5.25,0) to (4);
[thick, color=2] (5.25,0) to (5);
at (5,0.35) $Y_{I_{2}}$;
[thick, densely dashed, color = 2] (5) to (7.25,-0.5);
[thick, densely dashed, color = 2] (7.25,-0.5) to (6);
[thick, densely dashed, color = 2] (7.25,-0.5) to (7);
at (7,-0.9) ${I_2}$;

[thick, color = 6] (8) to (9);
at (11,0.35) $Y_{I_6}$;
[thick, densely dashed, color = 6,out=-45,in=-135] (8) to node[midway,below]${I_6}$ (11);
[thick, densely dashed, color = 3] (9) to (13.25,0);
[thick, densely dashed, color = 3] (13.25,0) to (10);
[thick, densely dashed, color = 3] (13.25,0) to (11);
at (13,0.35) ${I_3}$;
[thick, color = 3] (12) to (14.75,0);
[thick, color = 3] (14.75,0) to (10);
[thick, color = 3] (14.75,0) to (11);
at (15,0.35) $Y_{I_3}$;
(12) to (13);
[thick, densely dashed, color = 4] (12) to (14);
at (17,0.6) ${I_4}$;
(14) to (15);

Note that this diagram is disconnected and also violates Condition (<ref>), which requires that either both or neither of the pair of edges labeled by $e_1$ and $e_2$ in Figure <ref> must correspond to odd connections. In this example, only $e_1$ corresponds to an odd connection.

Let us return to considering the weakly rooted set $S = \{ 3, 4, 5 ,6, 8, 0 \}$ and maximal nested collection $\mathcal{I}$ from Figure <ref>. We previously exhibited the set of admissible matchings of $\mathcal{G}_{S}$ in Example <ref> and an example of a non-admissible matchings that violates Condition (<ref>) of Definition <ref> in Example <ref>.  The corresponding hyper $T$-path for this non-admissible matching is shown below. Odd connections are solid and even connections are dashed.

(1) at (-2,0) $8'$;
(2) at (0,0) $3$;
(3) at (2,0) $0'$;
(4) at (2,1) $9$;
(5) at (2,-1) $7'$;
(6) at (4,0) $4$;
(7) at (3.5,1.5) 0';
(8) at (4.5,1.5) 7;
(9) at (6,1) 9;
(10) at (6,0) 7';
(11) at (6,-1) 0';
[color=8,thick] (1) to (2);
[above] at (-1,0)$Y_{I_8}$;
[color=black!10!white,thick, line width = 10pt] (2) to (1.25,0);
[color=black!10!white,thick, line width = 10pt] (1.25,0) to (3);
[color=black!10!white,thick, line width = 10pt] (1.25,0) to (4);
[color=black!10!white,thick, line width = 10pt] (1.25,0) to (5);
[color=4,thick] (2) to (1.25,0);
[color=4,thick] (1.25,0) to (3);
[color=4,thick] (1.25,0) to (4);
[color=4,thick] (1.25,0) to (5);
[above] at (0.75,0)$Y_{I_4}$;
[color=black!10!white,thick, line width = 10pt] (3) to (6);
[color=black!10!white,thick, line width = 10pt] (5) to (2.75,0);
[color=7, densely dashed,thick] (3) to (6);
[color=7, densely dashed,thick] (5) to (2.75,0);
[below] at (3.25,0)${I_7}$;
[color=0,thick] (6) to (4,1);
[color=0,thick] (4,1) to (7);
[color=0,thick] (4,1) to (8);
[left] at (4,0.6)$Y_{I_0}$;
(6) to (9);
[color=black!10!white,thick, line width = 10pt] (6) to (10);
[color=black!10!white,thick, line width = 10pt] (11) to (5.25,0);
[color=7,thick] (6) to (10);
[color=7,thick] (11) to (5.25,0);
[below] at (4.75,0)$Y_{I_7}$;
[color=black!10!white,thick, line width = 10pt] (10) to (6.75,0);
[color=black!10!white,thick, line width = 10pt,out=0,in=90] (6.75,0) to (7,-1);
[color=black!10!white,thick, line width = 10pt,out=-90,in=180] (2) to (4,-2);
[color=black!10!white,thick, line width = 10pt, out=-90, in =0]  (7,-1) to (4,-2);
[color=black!10!white,thick, line width = 10pt] (9) to (6.75,0);
[color=black!10!white,thick, line width = 10pt] (11) to (6.75,0);
[color=4, densely dashed,thick] (10) to (6.75,0);
[color=4, densely dashed,out=0,in=90,thick] (6.75,0) to (7,-1);
[color=4, densely dashed,out=-90,in=180,thick] (2) to (4,-2);
[color=4, densely dashed, out=-90, in =0,thick]  (7,-1) to (4,-2);
[color=4, densely dashed,thick] (9) to (6.75,0);
[color=4, densely dashed,thick] (11) to (6.75,0);
[above] at (4,-2)${I_4}$;

(12) at (8.5,0) $1'$;
(13) at (10,0) $1$;
(14) at (11.5,0) $3$;
(15) at (13,0) $2$;
(12) to (13);
(14) to (15);
[color=4,thick,densely dashed] (13) to (14);
[above] at (10.75,0)$I_4$;

Note that this diagram both is disconnected and violates Condition (<ref>) of Definition <ref> due to the highlighted cycle, which uses connections $o_1, e_1, o_2, e_2$ such that $W(o_1) = W(e_1) = Y_{I_7}$ and $W(o_2) = W(e_2) = Y_{I_4}$. Hence, it is not a valid hyper $T$-path for $S$.

Next, we exhibit two hyper $T$-paths that do correspond to admissible matchings of $\mathcal{G}_{345680}$. We begin with the valid hyper $T$-path with weight $\frac{Y_{8}Y_{45670}Y_{560}^2}{Y_{12}Y_{2}Y_{5670}}$.

(0) at (12,0) $2$;
(1) at (10,0) $3$;
(2) at (8,0) $1$;
(3) at (6,0) $1'$;
(4) at (6,2) $3$;
(5) at (4,2) $8'$;
(6) at (8,2) $7'$;
(7) at (8,3) $9$;
(8) at (8,1) $0'$;
(9) at (10,2) $3$;
(10) at (8,4) $4$;
(11) at (6,4.5) $9$;
(12) at (10,3.5) $7'$;
(13) at (10,4.5) $0'$;
(14) at (12,3.5) $7$;
(15) at (12,4.5) $0'$;

[color=8, thick] (5) to (4);
[above] at (5,2)$Y_{I_8}$;
[color=1, thick, densely dashed] (4) to (3);
[left] at (6,1)${I_1}$;
[color=1, thick] (3) to (2);
[below] at (7,0)$Y_{I_1}$;
[color=2, thick, densely dashed] (2) to (1);
[below] at (9,0)${I_2}$;
(1) to (0);

[color=4, thick, densely dashed] (4) to (7.25,2);
[below] at (6.75,2)${I_4}$;
[color=4, thick, densely dashed] (6) to (7.25,2);
[color=4, thick, densely dashed] (7) to (7.25,2);
[color=4, thick, densely dashed] (8) to (7.25,2);

[below] at (9.25,2)$Y_{I_4}$;
[thick, color=4] (8.75,2) to (9);
[thick, color=4] (8.75,2) to (6);
[thick, color=4] (8.75,2) to (7);
[thick, color=4] (8.75,2) to (8);

[out = 60, in = 210] (4) to (10);
(10) to (11);
[color=7, thick, densely dashed] (10) to (9.25,4);
[color=7, thick, densely dashed] (12) to (9.25,4);
[color=7, thick, densely dashed] (13) to (9.25,4);
[below] at (9,4)${I_7}$;
[color=0, thick] (13) to (15);
[above] at (11,4.5)$Y_{I_0}^2$;
(12) to (14);

Note that the set of boundary nodes of the hyper $T$-path with weight $Y_{8}Y_{45670}Y_{0}^2$ is larger than $S' = \{2, 7, 8', 9, 0' \}$: it also contains a vertex labeled by $3$ because there is a vertex labeled 3 with valence $0 \oplus 1$ in $\mathcal{G}_{345680}$. This is allowed in our more general definition for hyper $T$-paths for weakly rooted sets, but was not allowed in the definition of hyper $T$-paths for rooted clusters in <cit.>. Additionally, we see that there is a vertex labeled 3 with  more than one adjacent even connection. In $\mathcal{G}_{345680}$, this vertex had valence $2 \oplus 0$ and therefore Condition (<ref>) of our more general definition for hyper $T$-paths requires that it be adjacent to exactly two even connections. This would not have been allowed by the definition in <cit.>.

Then we consider the valid hyper $T$-path with weight $\frac{Y_{123456780}Y_{45670}}{Y_{12}Y_{5670}}$.

(0) at (0,0) $8'$;
(1) at (2,-0.5) $0'$;
(2) at (2,-1.5) $1'$;
(3) at (2,0.5) $7'$;
(4) at (2,1.5) $9$;
(5) at (4,0) $3$;
(6) at (6,-1.5) $1$;
(7) at (6,0) $0'$;
(8) at (6,1) $7'$;
(9) at (6,2) $9$;
(10) at (8,0.5) $4$;
(11) at (10,1) $7$;
(12) at (10,0) $0'$;
(13) at (8,-1.5) $3$;
(14) at (10,-1.5) $2$;

[color=3,thick] (0) to (1.25,0);
[color=3,thick,out=0,in=135] (1.25,0) to (1);
[color=3,thick] (1.25,0) to (2);
[color=3,thick,out=0,in=-135] (1.25,0) to (3);
[color=3,thick] (1.25,0) to (4);
at (0.75,0.3) $Y_{I_3}$;

[color=4,thick,densely dashed] (5) to (2.75,0);
[color=4,thick,densely dashed,in=-70,out=150] (3,0) to (4);
[color=4,thick,densely dashed,out=180,in=45] (2.75,0) to (1);
[color=4,thick,densely dashed,out=180,in=-45] (2.75,0) to (3);
at (3.4,0.3) ${I_4}$;

[color=1,thick,densely dashed] (2) to (5);
at (3.25,-1) ${I_1}$;

[color=2,thick] (5) to (6);
at (4.6,-0.9) $Y_{I_7}$;
[color=2,thick,densely dashed] (6) to (13);
at (7,-1.85) ${I_2}$;
(13) to (14);

[color=4,thick] (5) to (8);
[color=4,thick] (5.3, 0.65) to (9);
[color=4,thick] (5.3, 0.65) to (7);
at (4.75,0.75) $Y_{I_4}$;

[color=7,thick,densely dashed] (10) to (6.75,0.5);
[color=7,thick,densely dashed] (6.75,0.5) to (8);
[color=7,thick,densely dashed] (6.75,0.5) to (7);
at (7.15,0.2) ${I_7}$;

[color=0,thick] (10) to (9.25,0.5);
[color=0,thick] (9.25,0.5) to (11);
[color=0,thick] (9.25,0.5) to (12);
at (8.85,0.2) $Y_{I_0}$;

Notice that this hyper $T$-path also contains a node (labeled by 3) that has multiple adjacent even connections.

Consider the graph and maximal nested collection from Example <ref>. The inadmissible matching shown in Figure <ref> would correspond to the (invalid) hyper $T$-path shown below.

(1) at (0,0) $1$;
(2) at (2,0) $0$;
(3) at (4,0) $2$;
(4) at (4,1.5) $0$;
(5) at (6,1) $4$;
(6) at (6,2) $3'$;
(7) at (8,1) $0$;
(8) at (10,1) $5'$;
(9) at (10,2) $3'$;
(10) at (10,0) $6'$;
(11) at (12,1) $0'$;

(1) to (2);
[thick, densely dashed, color=1] (2) to (3);
at (3,-0.4) ${I_1}$;
[thick, color=1] (3) to (4);
at (3.7,0.75) $Y_{I_1}$;
(3) to (5);
[thick, densely dashed,color=2] (4) to (5.25,1.5);
[thick, densely dashed,color=2] (5.25,1.5) to (6);
[thick, densely dashed,color=2] (5.25,1.5) to (5);
at (4.75,1.8) ${I_2}$;
[thick,color=2] (5) to (7);
[thick,color=2] (6) to (7,1);
at (7,0.65) $Y_{I_2}$;
[thick,color=4, densely dashed] (7) to (8);
[thick,color=4, densely dashed] (9.4,1) to (9);
[thick,color=4, densely dashed] (9.4,1) to (10);
at (9,1.25) $I_4$;
[thick, color=0] (8) to (11);
[thick, color=0] (9) to (10.6,1);
[thick,color=0] (10) to (10.6,1);
at (11, 1.25) $Y_{I_0}$;

From Example <ref>, we know that $\mathcal{G}_{0}$ contains internal edges $2 - 4$ and $2 - 3'$ such that an admissible matching of $\mathcal{G}_{0}$ must use either both or neither of these edges. Hence, a valid hyper $T$-path should contain odd connections that correspond to both or neither of these internal edges. The hyper $T$-path displayed above contains an odd connection that corresponds to the internal $2 - 4$ edge, but does not contain one that corresponds to the internal $2 - 3'$ edge, violating Condition (<ref>).

§ PROOF FOR SINGLETON SETS

This section and the following section consist of a proof of <Ref>.

We first recall some relevant notation from Section <ref> that will be used extensively in the next two sections:

* We have $N_{\Gamma'}(v)=\{a_1,\dots,a_p\}$ with $a_i<_{\mathcal I} v$ when $1\leq i\leq r$ and $a_i>_{\mathcal I} v$ when $r< i\leq p$ for some $r$.

* For $1\leq i\leq p$, we define $c_i = \vert \{I \in \mathcal{I} : I_{a_i} \subseteq I \subsetneq I_v\} \vert$.

* The sets counted by $c_i$ are $I^i_j$ such that $I_{a_i} = I^i_{c_i} \subset I^i_{c_i-1} \subset \cdots \subset I^i_1$.

* We have $a_j^i := m_{\mathcal{I}}(I_j^i)$ for $1\leq i\leq p$ and $1\leq j\leq c_i$.  In other words, $I^i_j=I_{a^i_j}$.  Notice that
$a_{c_i}^i=a_i$, and $\mathcal{C}_v = \{a_1^1,\ldots,a_1^r\}$.

In addition, we will define the following:

* $I_{c_i+1}^i := \emptyset$ for $1\leq i\leq p$

* For all $1 \leq i \leq r$,

\begin{align*}
A_j^i := \begin{cases} \bigg( \prod_{w \in \mathcal{C}_{a_{j}^i} \backslash \{a_{j+1}^i\}} Y_{I_w} \bigg) \cdot Y_{I_{j+1}^i - [v,a_j^i]}^2 & j < c_i \\
\prod_{w \in \mathcal{C}_{a_{c_i}^i}} Y_{I_w} & j = c_i \\
\end{cases}
\end{align*}

This section contains a proof of <Ref> for singleton sets; recall all singleton sets are weakly rooted.  In Section <ref>, we give an explicit formula for variables $Y_v$. In fact, we give a more general formula for $Y_{S}$ where $S = \{v\} \cup I_{m_1}^1(v) \cup \cdots \cup I_{m_r}^r(v)$ for values $1 \leq m_i \leq c_i+1$ by applying the exchange relations for graph LP algebras to a specific mutation sequence; recall the graphs $\mathcal{G}_S$ were constructed in Section <ref>. In Section <ref>, we prove Theorem <ref> for singleton sets $\{ v \}$ where $r = 1$ or $r=2$. In Section <ref>, we then prove Theorem <ref> for singleton sets $\{ v \}$ where $r > 2$.

§.§ Formulas in $\mathcal{A}_\Gamma$

When the rooted portion $\bar{S}$ of a set $S$ that is weakly rooted with respect to a maximal nested collection $\mathcal{I}$ is a singleton, we can compute $Y_{\bar{S}}$ directly by exhibiting a mutation sequence from $\mathcal{I}$ to another maximal nested collection that contains $\bar{S}$.

In the following proposition, we will  consider sets of the form  $J_{\mathbf{m}}(v) = \{v\} \cup I_{m_1}^1 \cup \cdots \cup I_{m_r}^r$ where $1 \leq m_i \leq c_i+1$ for each $i$.  When the vertex $v$ is understood, we will denote this set simply as $J_{\mathbf{m}}$. For convenience, we will assume $a_1,\dots,a_r$ are labeled such that $m_i > 1$ for $1 \leq i \leq t$ and $m_i = 1$ for $t+1 \leq i \leq r$ for some $0 \leq t \leq r$.

For any choice of $1 \leq m_i \leq c_i+1$ for each $1 \leq i \leq r$, let $J_{\mathbf{m}} =\{v\} \cup I_{m_1}^1 \cup \cdots \cup I_{m_r}^r$. Then,
\[
Y_{J_{\mathbf{m}}} = Y_{J_{\mathbf{m}} \backslash \{v\}} \bigg( \frac{Y_{I_v}}{Y_{I_1^1}\cdots Y_{I_1^r}} + \sum_{i=1}^r \sum_{j = 1}^{m_i-1} \frac{A_j^i}{Y_{I_j^i} Y_{I_{j+1}^i}} \bigg).
\]

If $t = 0$, $J_{\mathbf{m}}=I_v \in \mathcal{I}$, so this case is trivial.  For $t > 0$ we exhibit a mutation sequence that results in a cluster $\mathcal{I}'$ containing $J_{\mathbf{m}}$.  Namely, this sequence (written left-to-right) is given by mutating at $I_1^1 I_2^1, \dots, I_{m_1-1}^1, I_1^2, I_2^2,\dots,I_{m_t - 2}^t,  I_{m_t-1}^t$.

For $0 \leq i \leq t$, let $W^i = I_{m_1}^1 \cup I_{m_2}^2 \cup \cdots \cup I_{m_{i-1}}^{i-1} \cup I_1^{i+1} \cup \cdots \cup I_1^r$.  Define ${\widetilde I^i_j}:=W^{i} \cup I_{j+1}^i \cup \{v\}$ for $0\leq j \leq m_i$.  One can check that  $\widetilde I_{0}^{i+1} = \widetilde I_{m_i-1}^i$.  Observe that each mutation in this sequence replaces $I_j^i$ with ${\widetilde I^i_j}$. Each $\widetilde I_j^i$ is a connected vertex subset because every connected component of $W^{i}$ is adjacent to $v$, as is $I_{j+1}^{i}$.

Using Lemma 4.11 of <cit.>, we obtain exchange relations of the form

\begin{equation*}
Y_{\widetilde I^i_j}Y_{I^i_j}=Y_{\widetilde I^i_{j-1}}Y_{I^i_{j+1}} + Y_{W^{i}}\bigg(\prod_{w \in {\cal C}_{a_j^i} \backslash \{a_{j+1}^i\}} Y_{I_w}\bigg)Y_{I_{j+1}^i - [v,a_j^i]}^2 = Y_{\widetilde I^i_{j-1}}Y_{I^i_{j+1}} + Y_{W^{i}}A_j^i,
\end{equation*}

for $1 \leq i \leq t$ and $1 \leq j \leq m_i$.

The final mutation in our sequence produces the relation

$$Y_{\widetilde I^t_{m_t-1}}Y_{I^t_{m_t-1}}= Y_{\widetilde I^t_{m_t-2}}Y_{I^t_{m_t}} + Y_{W^{t}}A_{m_t-1}^t.$$

Because $Y_{\widetilde I^t_{m_t-1}}=Y_{J_{\bf m}}$, this can be equivalently written as

$$Y_{J_{\bf m}}= Y_{\widetilde I^t_{m_t-2}}\frac{Y_{I^t_{m_t}}}{Y_{I^t_{m_t-1}}} + \frac{Y_{W^{t}}}{Y_{I^t_{m_t-1}}}A_{m_t-1}^t.$$

We can then substitute the formula we have for $Y_{\widetilde I^t_{m_t-2}}$ from the previous mutation to obtain another formula for $Y_{J_{\bf m}}$, which contains $Y_{\widetilde I^t_{m_t-3}}$  if $m_t>2$ and $Y_{\widetilde I^{t-1}_{m_{t-1}-1}}$ if $m_t=2$. Via repeated recursive substitution using our formulas from previous mutations, we will eventually produce a formula for $Y_{J_{\bf m}}$ in terms the $A^i_j$'s and the variables in the original cluster.

Because every mutation has exactly one term containing $Y_{\widetilde I^i_j}$, we will eventually make the substitution
$$Y_{\widetilde I_1^1} = Y_{I_v} \frac{Y_{I_2^1}}{Y_{I_1^1}} + \frac{Y_{W^1}}{Y_{I_1^1}}A^1_1$$
in exactly one place. Hence, our formula for $Y_{J_{\bf m}}$ will have exactly one term containing $Y_{I_{v}}$, which we can compute as

$$Y_{I_v}\cdot\frac{Y_{I_2^1}}{Y_{I_1^1}} \cdot \frac{Y_{I_3^1}}{Y_{I_2^1}}\cdot \frac{Y_{I_4^1}}{Y_{I_3^1}} \cdots \frac{Y_{I_{m_1}^1}}{Y_{I_{m_1-1}^1}}\cdot \frac{Y_{I_2^2}}{Y_{I_1^2}}\cdot \cdots \cdot \frac{Y_{I_{m_t}^t}}{Y_{I_{m_t-1}^t}} = \frac{Y_{J_{\mathbf{m}} \backslash \{v\}} Y_{I_v}}{Y_{I_1^1}\cdots Y_{I_{1}^t}}.$$

For each mutation, the term that does not contain a factor $Y_{\widetilde I^i_j}$ is divisible by $A^i_j$.  This means our formula for $Y_{J_{\bf m}}$ will have exactly one term containing $A^i_j$ for $1\leq i\leq t$ and $1\leq j\leq m_i$.  Tracking this term through further substitutions, we can see that it has the form

\[
\frac{Y_{W^{i}}A_j^i}{Y_{I_j^i}} \cdot \frac{Y_{I_{j+2}^i}}{Y_{I_{j+1}^i}}\cdot\frac{Y_{I_{j+3}^i}}{Y_{I_{j+2}^i}}\cdots \frac{Y_{I_{m_i}^i}}{Y_{I_{m_i-1}^i}}\cdot\frac{Y_{I_2^{i+1}}}{Y_{I_1^{i+1}}}\cdot\frac{Y_{I_3^{i+1}}}{Y_{I_2^{i+1}}}\cdots  \frac{Y_{I_{m_t}^t}}{Y_{I_{m_t-1}^t}} = \frac{ Y_{J_{\mathbf{m}} \backslash \{v\}} A_i^j }{Y_{I_j^i}Y_{I_{j+1}^i}},
\]

using the fact that $Y_{W^{i+1}}$ has factors $ Y_{I_1^{i+1}} Y_{I_1^{i+2}} \cdots Y_{I_1^r}$, which cancel with factors in the denominator.

This accounts for all terms in this sequence of mutations, completing the proof.

By setting all $m_i = c_i+1$, we obtain a formula for variables of the form $Y_{\{v\}}$.

Let $v \in V(\Gamma)$. Then
\[
Y_{\{v\}} = \frac{Y_{I_v}}{Y_{I_1^1}\cdots Y_{I_1^r}} + \sum_{i = 1}^r \sum_{j = 1}^{c_i} \frac{A_j^i}{Y_{I_{j}^i} Y_{I_{j+1}^i}}
\]

where $I_{c_i+1}^i := \emptyset$.

It will be convenient to use the shorthand $Z(v) := \frac{Y_{I_v}}{\prod_{i \in \mathcal{C}_v} Y_{I_i}}$ for $v \in V(\Gamma)$. Using this notation, we can rewrite the statement of Corollary <ref> as $Y_{\{v\}} = Z(v) + \sum_{i=1}^r \sum_{j=1}^{c_i} \frac{A_j^i}{Y_{I_j^i}Y_{I_{j+1}^i}}$.

Consider the nested collection from our running example in <Ref>.  By applying Proposition <ref> with $v=4$ and $m_1 = 2$, we can compute $Y_{4560}$ as

\begin{align*}
Y_{\{4\}\cup I^1_2} &= Y_{I^1_2}\left(\frac{Y_{I_4}}{Y_{I^1_1}}+\frac{A^1_1}{Y_{I^1_1}Y_{I^1_2}}\right)\\
Y_{\{4\}\cup I^1_2} &= Y_{I^1_2}\left(\frac{Y_{I_4}}{Y_{I^1_1}}+\frac{(\prod_{w\in\mathcal{C}_7\setminus\{0\}}Y_{I_w})Y^2_{I^1_2\setminus[4,7]}}{Y_{I^1_1}Y_{I^1_2}}\right)\\
Y_{4560} &= Y_{560}\left(\frac{Y_{45670}}{Y_{5670}}+\frac{Y^2_0}{Y_{5670}Y_{560}}\right)\\
Y_{4560} &= \frac{Y_{45670}Y_{560}+Y^2_0}{Y_{5670}}\\
\end{align*}

By Corollary <ref>, we can write $Y_4$ as

\begin{align*}
Y_4 &= \frac{Y_{I_4}}{Y_{I_1^1
}} + \frac{A^1_1}{Y_{I^1_1}Y_{I^1_2}} + \frac{A^1_2}{Y_{I^1_2}Y_{I^1_3}} + \frac{A^1_3}{Y_{I^1_3}Y_{\emptyset}}\\
&= \frac{Y_{I_4}}{Y_{I_1^1
}} + \frac{(\prod_{w\in\mathcal{C}_7\setminus\{0\}}Y_{I_w})Y^2_{I^1_2\setminus[4,7]}}{Y_{I^1_1}Y_{I^1_2}} + \frac{(\prod_{w\in\mathcal{C}_0\setminus\{5\}}Y_{I_w})Y^2_{I^1_3\setminus[4,0]}}{Y_{I^1_2}Y_{I^1_3}} + \frac{\prod_{w\in\mathcal{C}_5}Y_{I_w}}{Y_{I^1_3}Y_{\emptyset}}\\
&= \frac{Y_{45670}}{Y_{5670
}} + \frac{Y^2_0}{Y_{5670}Y_{560}} + \frac{Y^2_6}{Y_{560}Y_{56}} + \frac{Y_6}{Y_{56}}
\end{align*}

§.§ r = 1, 2

We first prove our main theorem is true for snake graphs $\mathcal{G}_v$ when $r=1$ or $r=2$.
Following <Ref>, we have $A := \{\ell\in \Gamma'\setminus \Gamma\ |\ a_1\in [v,l]\}$, $B := \{k\in \Gamma'\setminus \Gamma\ |\  a_1\notin [v,k]\}$, and $b_{i,j}^{\ell,k}$ denotes the vertex other than $v$ in  $N_{\Gamma'}(I_j^i) \cap [\ell,k]$.

When $r=1$ or $r=2$, in $\cal G_v$:

* The vertex $v_\opt$ has valence $1\oplus p-2$.

* The vertex $a_i$ has valence $1\oplus \deg_{\Gamma'}(a_i) - 2$ for $1\leq i\leq r$.

* The vertex $a_j^i$ incident to diagonal $I_{j+1}^i$ has valence $1\oplus \deg_{\Gamma'}(a_j^i) - 2$ for each $1\leq i\leq r$ and $1\leq j\leq c_i$.

* The rest of the vertices are either labeled $v$, $a_j$ for some $r< j\leq p$, or $b_{i,j}^{\ell,k}$ for some $\ell\in A, k\in B, 1\leq i\leq r$, and $1\leq j\leq c_i$.  These vertices all have valence $1\oplus 0$.

Recall that the component snake graphs $H_{\ell,k}$ have the form shown in <Ref> or in <Ref>.

First we consider the vertices in $\cal G_v$ with label $v$ except for $v_\opt$. Each vertex $v$ except for the vertex $v_{\text{opt}}$ is incident to edges with the same weights in each component snake graph.  We know that as we glue the component snake graphs to form  $\cal G_v$, these edges with the same weights will be glued into one hyperedge.  Thus, each corresponding vertex in $\mathcal{G}_v$ will have valence $1 \oplus 0$.

We then consider the vertex $v_{\opt}$, which has degree two in each of the component snake graph. One of these edges always has weight $Y_{I^1_2}$.  In the process of creating $\mathcal{G}_v$, these will get combined.  If $r=2$, we will also combine the edges $\{(v_{\opt},b_{i,1}^{\ell,k}):\ell\in A,k\in B, 2\leq i\leq r\}$ which will create a single hyperedge with label $I^2_1$.  For both $r=1$ and $r=2$, there will also be component snake graphs with edges from the set $\{(v_\opt,a_i):r < i\leq p\}$ which will not be identified for different $i$, so that $v_\opt$ is incident to $p$ edges in $\cal G_v$. Thus, we see that $v_\opt$ has valence $1\oplus (p-2)$ in $\mathcal{G}_v$. The vertices $a_i$ for $r < i \leq p$ are always incident to these edges as well as the edge $Y_{I_v}$, so they will have valence $1\oplus 0$.

Next, we look at what happens to a vertex $b_{1,j}^{\ell,k}$ from $H_{\ell,k}$.  First suppose $1<j<c_1$.  Then the vertex $b_{1,j}^{\ell,k}$  in $H_{\ell,k}$ will be incident to two different vertices $v$, a vertex $b_{1,j-1}^{\ell,k}$, and a vertex $b_{1,j+1}^{\ell,k}$.  Consider some other $H_{\ell',k'}$ where $b_{1,j}^{\ell',k'}=b_{1,j}^{\ell,k}$. This vertex will also have two edges incident to two different vertices $v$.  These will get  identified with the corresponding edges from $H_{\ell,k}$ when we create $\mathcal{G}_v$.  For all $\ell',k'$ with $b_{1,j}^{\ell',k'}=b_{1,j}^{\ell,k}$, we will have $b_{1,j+1}^{\ell',k'}=b_{1,j+1}^{\ell,k}$ because $N_{\Gamma'}(I^1_{j+1})\cap[\ell,k]=N_{\Gamma'}(I^1_{j+1})\cap[b_{1,j}^{\ell,k},v]=N_{\Gamma'}(I^1_{j+1})\cap[\ell',k']$.  This means that the edges $(b_{1,j}^{\ell,k}, b_{1,j+1}^{\ell,k})$ and $(b_{1,j}^{\ell',k'}, b_{1,j+1}^{\ell',k'})$ will also get identified when constructing  $\mathcal{G}_v$.  Thus, any additional valence comes from edges $(b_{1,j}^{\ell',k'}, b_{1,j-1}^{\ell',k'})$.

If $b_{1,j}^{\ell,k}\in I^1_{j-1}$, then $b_{1,j}^{\ell,k}=a_{j-1}^1$. Since the path $[\ell',k']$ passes through $a_{j-1}^1$, it will pass through two neighbors of $a_{j-1}^1$: the unique neighbor in $[a_{j-1}^1,v]$ and another. This second neighbor, call it $w$, is either in $I_{j-1}^1$ or not. If $w \in I_{j-1}^1$, then $w$ is contained in a set $I_z$ for $z \in \mathcal{C}_{a_{j-1}^1}$ and the path $[\ell',k']$ will pass through an element of $N_{\Gamma'}(I_z) \backslash \{a_{j-1}^1\}$. Thus, the edge $(b_{1,j-1}^{\ell',k'},b_{1,j}^{\ell',k'}) = (b_{1,j-1}^{\ell',k'},a_{j-1}^1)$ will have weight $Y_{I_z}$. Otherwise, $w$ is not in $I_{j-1}^1$. In this case, it must be that $(b_{1,j-1}^{\ell',k'},b_{1,j}^{\ell',k'})=(w,a_{j-1}^1) \in E(\Gamma)$, so we weight this edge with 1. Notice that in $\mathcal{G}_v$, all edges with weight $Y_z$  will get combined for each $z$ and the edges of weight 1 will all remain separate edges.  This means that there will be $\deg_{\Gamma'}(a_{j-1}^1)-1$ edges adjacent to the vertex $a_{j-1}^1$ in $\mathcal{G}_v$ that came from edges $(b_{1,j}^{\ell',k'}, b_{1,j-1}^{\ell',k'})$ with $b_{1,j}^{\ell',k'}=a_{j-1}^1$.  This gives the vertex $a_{j-1}^1$ a valence of $1\oplus\deg_{\Gamma'}(a_{j-1}^1)-2$.

If $b_{1,j}^{\ell,k}\not\in I^1_{j-1}$, then $b_{1,j}^{\ell,k}\in N_{\Gamma'}(I^1_{j-1})$.  Notably, in this case $b_{1,j}^{\ell,k}\neq a_{j-1}^1$ and $b_{1,j}^{\ell,k} = b_{1,j-1}^{\ell,k}$. Since the vertex $b_{1,j+1}^{\ell,k}$ is also determined by $b_{1,j}^{\ell,k}$, we know this vertex is incident to the same edges in all component snake graphs where it appears, so $b_{1,j}^{\ell,k}$ will have valence $1\oplus0$ in $\mathcal{G}_v$.

Now consider the vertex $b_{1,c_1}^{\ell,k}$ in $H_{\ell,k}$.  If $c_1=1$ then $b_{1,c_1}^{\ell,k}$ will either be incident to a vertex $a_1$ and a vertex $a_i$ for some $r<i$ or it will be incident to a vertex $a_1$, a vertex $b_{i,1}^{\ell,k}$ for some $1<i\leq r$, and a vertex $v$.  Consider some other $H_{\ell',k'}$ where $b_{1,c_1}^{\ell',k'}=b_{1,c_1}^{\ell,k}$. This vertex will either have edges incident to a vertex $a_1$ and a vertex $a_{i'}$ for some $r<i'$ or it will have edges incident to a vertex $a_1$, a vertex $b_{i',1}^{\ell',k'}$ for some $1<i'\leq r$, and a vertex $v$.  Note that the edge from $b_{1,c_1}^{\ell,k}$ to $a_i$ or $b_{i,1}^{\ell,k}$ and the edge from $b_{1,c_1}^{\ell',k'}$ to $a_{i'}$ or $b_{i',1}^{\ell',k'}$ will both have weight $Y_{I_v}$.  Thus the edges extending from $b_{1,c_1}^{\ell',k'}$ in each direction will get identified with the corresponding edges from $H_{\ell,k}$ when we create $\mathcal{G}_v$. So $b_{1,c_1}^{\ell,k}$ will have valence $1\oplus 0$. Note that in this case $b_{1,c_1}^{\ell,k}\neq a^i_j$ for any $i,j$ as $b_{1,c_1}^{\ell,k}=b_{1,1}^{\ell,k}\not\in I_1^i$ for any $i$.  If $c_1\neq 1$, the vertex $b_{1,c_1}^{\ell,k}$ in $H_{\ell,k}$ will be incident to a vertex $v$, a vertex $a_1$, and a vertex $b_{1,c_1-1}^{\ell,k}$.  Consider some other $H_{\ell',k'}$ where $b_{1,c_1}^{\ell',k'}=b_{1,c_1}^{\ell,k}$. This vertex will have edges incident to a vertex $v$ and a vertex $a_1$ which will get identified with the corresponding edges from $H_{\ell,k}$ when we  create $\mathcal{G}_v$.  The logic from case of $b_{i,j}^{\ell,k}$ with $1<j<c_1$ applies here to obtain our conclusion.

Our next case is vertices $b_{1,1}^{\ell,k}$.  Consider the vertex $b_{1,1}^{\ell,k}$ in $H_{\ell,k}$.  If $c_1=1$ we are done by a previous case.  Otherwise, this vertex will be incident to two vertices $v$, a vertex $b_{1,2}^{\ell,k}$, and either a vertex $a_i$ for some $i>r$ or a vertex $b_{i,1}^{\ell,k}$ for some $1<i\leq r$.  Consider some other $H_{\ell',k'}$ where $b_{1,1}^{\ell',k'}=b_{1,1}^{\ell,k}$. This vertex will have edges incident to two vertices $v$, a vertex $b_{1,2}^{\ell',k'}$, and either a vertex $a_{i'}$ for some $i'>r$ or a vertex $b_{i',1}^{\ell',k'}$ for some $1<i'\leq r$.  The edge from $b_{1,1}^{\ell,k}$ to $a_i$ or $b_{i,1}^{\ell,k}$ and the edge from $b_{1,1}^{\ell',k'}$ to $a_{i'}$ or $b_{i',1}^{\ell',k'}$ will both have weight $Y_{I_v}$, so they will get identified when we create $\mathcal{G}_v$. The edges adjacent to $b_{1,1}^{\ell,k}$ and vertices $v$ are exactly the same as edges adjacent to $b_{1,1}^{\ell',k'}$, and vertices $v$ so they will also get identified when we create $\mathcal{G}_v$.  For all $\ell',k'$ with $b_{1,1}^{\ell',k'}=b_{1,1}^{\ell,k}$, we will have $b_{1,2}^{\ell',k'}=b_{1,2}^{\ell,k}$ because $N_{\Gamma'}(I^1_{2})\cap[\ell,k]=N_{\Gamma'}(I^1_{2})\cap[b_{1,1}^{\ell,k},v]=N_{\Gamma'}(I^1_{2})\cap[\ell',k']$.  This means that the edges $(b_{1,1}^{\ell,k}, b_{1,2}^{\ell,k})$ and $(b_{1,1}^{\ell',k'}, b_{1,2}^{\ell',k'})$ will also get identified when constructing  $\mathcal{G}_v$. Thus, the  valence of $b_{1,1}^{\ell,k}$ in $\mathcal{G}_v$ is $1\oplus 0$.  As earlier, $b_{1,1}^{\ell,k}\neq a^i_j$ for any $i,j$ as $b_{1,1}^{\ell,k}\not\in I_j^i$ for any $i,j$.

The argument for vertices $b_{i,j}^{\ell,k}$ for $1<i\leq r$ and $1\leq j\leq c_i$ is exactly the same as in the cases where $i=1$ detailed above.

Finally, we analyze the vertex $a_i$ for $1\leq i\leq r$. In $\cal G_v$, the vertex $a_1$ is incident to an edge of the form $(a_1,v)$ which is the result of combining such an edge from every component snake graph. On the other hand, after gluing the edges $(a_1,b^{\ell,k}_{1,c_1})$, for all $\ell\in A$ and $k\in B$, the vertex $a_1$ will have $\deg_{\Gamma'}(a_1)$ more incident edges by the same logic as above for the $a_j^1$ vertices. Therefore $a_1$ has valence $1\oplus \deg_{\Gamma'}({a_1})-1$.  The same logic applies to $a_2$ if $r=2$.

Combining Lemma <ref> and Condition (<ref>) in Definition <ref> will give us a useful characterization of subsets of edges which must appear together in an admissible matching. We define $B_j^i = N_{\Gamma'}(I_j^i) \backslash \{v\}$.

Let $P$ be an admissible matching of $\mathcal{G}_v$ where $r \leq 2$. Let $e_1$ and $e_2$ be two internal edges which both share vertices with diagonals $d_1$ and $d_2$. Then, $e_1$ and $e_2$ are either both in $P$ or neither are in $P$.

The possible pairs of diagonals which can share vertices with the same edges would either have labels $d_1 = I_1^1$ and $d_2 = I_1^2$ or $d_1 = I_j^i$ and $d_2 = I_{j+1}^i$ for $1\leq i \leq r$ and $1 \leq j < c_i$. For the first case, note that there is exactly one internal edge, weighted $Y_{I_v}$, which shares vertices with diagonals $I_1^1$ and $I_1^2$, and so there is nothing to check in this case.

Consider the set of all internal edges which share vertices with diagonals $I_{j+1}^i$ and $I_j^i$ for some $1 \leq i \leq r$ and $1 \leq j < c_i$.  From step (4) in Section <ref>, we see that there is a set of edges which are each incident to $a_j^i \in B_{j+1}^i$ and that the remaining edges do not share vertices with this set or with each other. Condition (<ref>) of Definition <ref> implies that an admissible matching must include all of the former set of edges or none of them. By Lemma <ref>, if we include this entire set of edges, we cannot include any other edge incident to $a_j^i \in B_{j+1}^i$.

Suppose for sake of contradiction that there exists an admissible matching, $P$, which includes a proper subset of the set of internal edges which share vertices with diagonals $I_{j+1}^i$ and $I_j^i$. Suppose first that $j = c_i-1$. We know there are vertices in $B_{j+1}^i$ which are not covered by these internal edges, so they must be covered by some other edges. There is a hyperedge $Y_{I_{c_i}^i}$ incident to all vertices $B_{j+1}^i$. However, including this hyperedge which violate the valence of the edges which are incident to internal edges in $P$, so we cannot include it. There is also a set of edges from $a_i$, with labels $Y_{I_z}$ for $z \in \mathcal{C}_{a_i}$. Condition (<ref>) of Definition <ref> implies that an admissible matching must include all or none of this set, so we also cannot cover the vertices with these edges. Therefore, in this case, this configuration cannot be part of an admissible matching.

Now, let $j < c_i-1$. If we take a proper subset of the internal edges incident to diagonals $I_{j+1}^i$ and $I_j^i$, we again have vertices in $B_{j+1}^i$ which must be covered by other edges in $P$. As before, we cannot use either of the boundary edges incident to $B_{j+1}^i$ since this would violate the valence of vertices incident to this proper subset of internal edges. There is a second set of internal edges incident to $B_{j+1}^i$ in this case, namely the set which is incident to diagonals $I_{j+2}^i$ and $I_{j+1}^i$. It is possible that we can cover the necessary vertices in $B_{j+1}^i$ using these internal edges. However, since we cannot use all of these internal edges, we have now included a proper subset of these internal edges. Proceeding in this way, we would be required to use a proper subset of the internal edges which share vertices with diagonals $I_{c_i}^i$ and $I_{c_i-1}^i$, and by our previous paragraph this cannot be a subset of an admissible matching.

Let $B^i_j:= N_{\Gamma'}(I^i_j)\setminus\{v\}$ and $\tilde{a}:=\{a_{r+1},\dots,a_p\}$. If $r=1$ the admissible matchings of $\mathcal{G}_v$ are in weight preserving bijection with perfect matchings of the snake graph $\tilde{\mathcal{G}}_v$ shown in <Ref>, and if $r=2$ the admissible matchings of $\mathcal{G}_v$ are in weight preserving bijection with perfect matchings of the snake graph $\tilde{\mathcal{G}}_v$ shown in <Ref>.

(11) at (1,0) $v$;
(12) at (0,2) $v$;
(13) at (2,1) $v$;
(14) at (1,3) $v$;
(15) at (4,3) $v$;
(16) at (3,5) $v$;
(17) at (5,4) $v_{\opt}$;
(2) at (0,0) $a_1$;
(a) at (0,1) $B^1_{c_1}$;
(b) at (1,1) $B^1_{c_1-1}$;
(c) at (1,2) $B^1_{c_1-2}$;
(d) at (2,2) $B^1_{c_1-3}$;
(i) at (3,4) $B^1_3$;
(j) at (4,4) $B^1_2$;
(r) at (4,5) $B^1_1$;
(s) at (5,5) $\tilde{a}$;
at (2.5,3) $\reflectbox{$\ddots$}$;
(11) to (2);
(2) to node[scale=0.85,left]$A^1_{c_1}$ (a);
(a) to node[scale=0.85,above]$A^1_{c_1-1}$ (b);
(b) to node[scale=0.85,right]$A^1_{c_1-2}$ (c);
(c) to node[scale=0.85,above]$A^1_{c_1-3}$ (d);
(i) to node[scale=0.85,above]$A^1_2$ (j);
(r) to node[scale=0.85,right]$A^1_1$ (j);
(r) to node[scale=0.85,above]$Y_{I_v}$ (s);
(s) to (17);
(a) to node[scale=0.85,left]$Y_{I_{c_1}^1}$ (12);
(b) to node[scale=0.85,right]$Y_{I_{c_1-1}^1}$ (11);
(b) to node[scale=0.85,below]$Y_{I_{c_1-1}^1}$ (13);
(c) to node[scale=0.85,above]$Y_{I_{c_1-2}^1}$ (12);
(c) to node[scale=0.85,left]$Y_{I_{c_1-2}^1}$ (14);
(d) to node[scale=0.85,right]$Y_{I_{c_1-3}^1}$ (13);
(d) to node[scale=0.85,below]$Y_{I_{c_1-3}^1}$ (3,2);
(i) to node[scale=0.85,above]$Y_{I_3^1}$ (2,4);
(i) to node[scale=0.85,left]$Y_{I_3^1}$ (16);
(j) to node[scale=0.85,right]$Y_{I_2^1}$ (15);
(j) to node[scale=0.85,below]$Y_{I_2^1}$ (17);
(r) to node[scale=0.85,above]$Y_{I_1^1}$ (16);
[dashed] (11) to node[scale=0.85,above]$I_{c_1}^1$ (a);
[dashed] (b) to node[scale=0.85,above, xshift = 6]$I_{c_1-1}^1$ (12);
[dashed] (13) to node[scale=0.85,above, xshift = 6]$I_{c_1-2}^1$ (c);
[dashed] (d) to node[scale=0.85,above,xshift = 6]$I_{c_1-3}^1$ (14);
[dashed] (15) to node[scale=0.85,above]$I_3^1$ (i);
[dashed] (j) to node[scale=0.85,above]$I_2^1$ (16);
[dashed] (17) to node[scale=0.85,above]$I_1^1$ (r);

The snake graph $\tilde{\cal G}_v$ when $r=1$.

(11) at (1,0) $v$;
(12) at (0,2) $v$;
(13) at (2,1) $v$;
(14) at (1,3) $v$;
(15) at (4,3) $v_\opt$;
(16) at (3,5) $v_\fix$;
(17) at (6,5) $v$;
(2) at (0,0) $a_1$;
(a) at (0,1) $B_{c_1}^1$;
(b) at (1,1) $B_{c_1-1}^1$;
(c) at (1,2) $B_{c_1-2}^1$;
(d) at (2,2) $B_{c_1-3}^1$;
(i) at (3,4) $B_1^1$;
(j) at (4,4) $B_1^2\cup\tilde{a}$;
(r) at (5,6) $B_{c_2}^2$;
(s) at (6,6) $a_2$;
at (2.5,3) $\reflectbox{$\ddots$}$;
at (4.5,5) $\reflectbox{$\ddots$}$;
(11) to (2);
(2) to node[scale=0.85,left]$A_{c_1}^1$ (a);
(a) to node[scale=0.85,above]$A_{c_1-1}^1$ (b);
(b) to node[scale=0.85,right]$A_{c_1-2}^1$ (c);
(c) to node[scale=0.85,above]$A_{c_1-3}^1$  (d);
(i) to node[scale=0.85,above]$Y_{I_v}$ (j);
(r) to node[scale=0.85,above]$A_{c_2}^2$ (s);
(s) to (17);
(a) to node[scale=0.85,left]$Y_{I_{c_1}^1}$ (12);
(b) to node[scale=0.85,right]$Y_{I_{c_1-1}^1}$ (11);
(b) to node[scale=0.85,below]$Y_{I_{c_1-1}^1}$ (13);
(c) to node[scale=0.85,above]$Y_{I_{c_1-2}^1}$ (12);
(c) to node[scale=0.85,left]$Y_{I_{c_1-2}^1}$ (14);
(d) to node[scale=0.85,right]$Y_{I_{c_1-3}^1}$ (13);
(d) to node[scale=0.85,below]$Y_{I_{c_1-3}^1}$ (3,2);
(i) to node[scale=0.85,above]$Y_{I_1^1}$ (2,4);
(i) to node[scale=0.85,left]$Y_{I_1^1}$ (16);
(j) to node[scale=0.85,right]$Y_{I_1^2}$ (15);
(j) to node[scale=0.85,below]$Y_{I_1^2}$ (5,4);
(r) to node[scale=0.85,above]$Y_{I_{c_2}^2}$ (4,6);
[dashed] (11) to node[scale=0.85,above]$I_{c_1}^1$ (a);
[dashed] (b) to node[scale=0.85,above, xshift = 6]$I_{c_1-1}^1$ (12);
[dashed] (13) to node[scale=0.85,above, xshift = 6]$I_{c_1-2}^1$ (c);
[dashed] (d) to node[scale=0.85,above,xshift = 6]$I_{c_1-3}^1$ (14);
[dashed] (15) to node[scale=0.85,above]$I_1^1$ (i);
[dashed] (j) to node[scale=0.85,above]$I_1^2$ (16);
[dashed] (17) to node[scale=0.85,above]$I_{c_2}^2$ (r);

The snake graph $\tilde{\cal G}_v$ when $r=2$.

We begin discussing the structures of an admissible matching around each vertex in $\tilde{\cal{G}}_v$.

First of all, if the vertex $v$ adjacent to $a_1$ is not $v_\opt$, it is connected to an edge with weight $Y_{I^1_{c_1-1}}$ and an edge of weight 1.  An admissible matching of $\cal{G}_v$ will use exactly one of these edges, which corresponds to one of the two edges $(v,B^1_{c_1-1})$ and $(v,a_1)$ in $\tilde{\cal G}_v$. When $r=2$ the same is true for the vertex $v$ adjacent to $a_2$ as long as it is not $v_\fix$.  Similarly, every other vertex $v$ except for $v_\opt$ and $v_\fix$ is connected to two (hyper)edges with weights $Y_{I_{j+1}^i}$ and $Y_{I_{j-1}^i}$ respectively. An admissible matching of $\cal{G}_v$ will use exactly one of these edges, which corresponds to one of the two edges $(v,B^i_{j+1})$ and $(v,B^i_{j-1})$ in $\tilde{\cal G}_v$.

The vertex $v_\opt$ has higher valence, but it still has only two possibilities in $\cal{G}_v$. The vertices $a_j, j > r$ have degree 2 in $\cal{G}_v$ and if $r=2$ then the vertices in $B_1^2$ also have degree 2 in $\cal{G}_v$.  These vertices are each adjacent to the edge weighted $Y_{I_v}$ and an edge $(a_j,v_\opt)$ with weight 1 or the edge $(B_1^2,v_\opt)$ with weight $Y_1^2$. If the edge $Y_{I_v}$ is included in an admissible matching, then none of the edges $(a_j,v_\opt)$ or $(B_1^2,v_\opt)$ can be included in the matching. Conversely, if the edge $Y_{I_v}$ is not included in the matching, then we must use all the edges $(a_j,v_\opt)$ and edge $(B_1^2,v_\opt)$. This matches the configuration around $v_\opt$ in the ordinary snake graph $\tilde{\cal{G}_v}$.

If $r=1$, there is no vertex $v_\fix$.  If $r=2$, the vertex $v_\fix$ has valence $1\oplus 0$ and is connected to an edge of weight $Y_{I_1^1}$ and an edge of weight $Y_{I_2^2}$ respectively. An admissible matching of $\cal{G}_v$ will use exactly one of these edges, which corresponds to one of the two edges $(v,B^1_1)$ and $(v,B^2_2)$ in $\tilde{\cal G}_v$.

The vertex $a_1$ also has two possibilities in an admissible matching of $\cal G_v$. By Condition (<ref>) in Definition <ref>, if we include any of the edges between $a_1$ and vertices in $B_{c_1}^1)$ then we include all such edges.  Otherwise we instead must include $(a_1,v)$. This corresponds to the two choices we have for the vertex $a_1$ in $\tilde{\cal{G}_v}$.  When $r=2$, the same argument holds for the vertex $a_2$.

We now consider the vertices $a_i$ for $r<i\leq p$. Each such $a_i$ is connected to $v_\opt$ with an edge of weight 1 and connected to all the vertices in $B_1^j$ for all $1\leq j\leq r$ by an edge of weight $Y_{I_v}$.  Since the vertices $a_i$ all have valence $1\oplus 0$, every admissible matching either contains the edge $Y_{I_v}$ or all the edges $a_i$ to $v_\opt$.  This matches the configuration around $\tilde{a}$ in $\tilde{\cal G}_v$ for $r=1$.  For $r=2$, if the edge $Y_{I_v}$ is used in an admissible matching on $\tilde{\cal G}_v$ the edge $B_1^2$ to $v_\opt$ must not be used and if the edge $Y_{I_v}$ is not used in an admissible matching on $\tilde{\cal G}_v$, then the edge $B_1^2$ to $v_\opt$ must also be used. So the edge $B_1^2$ to $v_\opt$ is used exactly when the edges $a_i$ to $v_\opt$ are used.  This matches the configuration around $\tilde{a}$ in $\tilde{\cal G}_v$ for $r=2$.

We next discuss what happens around vertices $b_{i,j}^{\ell,k}\in B_j^i$ for some $1\leq i\leq r$ and $1\leq j\leq c_i$.  First suppose $b_{i,j}^{\ell,k}\not\in B_{c_1}^1,B_1^1,B_1^2,B_{c_2}^2$.  Then the relevant part of $\cal G_v$ will look as follows, where the sets $B_{j-1}^i,B_j^i$, and $B_{j+1}^i$ represent (possibly) multiple vertices and the double edges mean that there are (possibly) multiple hyperedges between vertices in the sets. By abusing notation, we use $(B_{j-1}^i,B_j^i)$ and $(B_j^i,B_{j+1}^i)$ to denote the set of edges between vertices in two sets in $\cal G_v$. Note that the vertices $v,v_1,$ and $v_2$ in the picture are all labeled by the same vertex $v$ from $\Gamma$, but we label them here so that they are distinguishable on $\cal G_v$.

(vopt) at (0,0)$v_2$;
(b12) at (1,0)$B_{j}^i$;
(b22) at (2,0)$B_{j-1}^i$;
(v) at (1,1)$v_1$;
(vfix) at (2,-1)$v$;
(b11) at (1,-1)$B_{j+1}^i$;
(vopt) to node[above]$Y_{I_j^i}$ (b12);
[double]  (b12) to node[above]$A_{j-1}^i$ (b22);
(b12) to node[left]$Y_{I_j^i}$ (v);
[double] (b12) to node[right]$A_j^i$ (b11);
(vfix) to node[right]$Y_{I_{j-1}^i}$ (b22);
[dashed] (v) to node[right]$I_{j-1}^i$ (b22);
[dashed] (b12) to node[right]$I_j^i$ (vfix);
(vfix) to node[below]$Y_{I_{j+1}^i}$ (b11);
[dashed] (vopt) to node[right]$I_{j+1}^i$(b11);

By Proposition <ref>, all edges in $(B_{j-1}^i,B_j^i)$ have to appear together in an admissible matching of $\cal G_v$, and similarly for $(B_j^i,B_{j+1}^i)$. We will show an admissible matching of $\cal G_v$ cannot contain any configuration near $B_j^i$ which would not correspond to a configuration near such a vertex in the ordinary snake graph $\tilde{\cal{G}_v}$.

* Since $a^i_{j-1}$ is a neighbor of $I_j^i$ other than $v$, $a^i_{j-1}\in B_j^i$.  From the proof of <Ref>, there are $\deg_{\Gamma'}(a^i_{j-1})-1$ edges from $a^i_{j-1}$ to $B_{j-1}^i$. Thus by <Ref>, if we use the edges in $(B_j^i,B_{j-1}^i)$ in an admissible matching, we can't include the edges in $(B_j^i,B_{j+1}^i)$ the edge $(v_1,B_j^i)$, or the edge $(v_2,B_j^i)$, as this would exceed the valence of $a_{j-1}^i\in B_j^i$.

* The edges $(v_1,{B_j^i})$ and $(v_2,{B_j^i})$ cannot be used together in an admissible matching because $v$ has valence $1\oplus 0$ by <Ref> and adding in either edge adjacent to $v$ would cause a violation of Condition (<ref>) of Definition <ref>.

* Suppose we use the edges $(B_j^i, B_{j+1}^i)$ as well as the edge $(v_1, B_j^i)$ in an admissible matching. Then by Condition (<ref>) of Definition <ref>, we cannot use the edge $(v, B_{j-1}^i)$. This means we must use the edge $(v, B_{j+1}^i)$. By the same argument as in part (a), we know $a^i_{j}\in B_{j+1}^i$ and there are $\deg_{\Gamma'}(a^i_{j})-1$ edges from $a^i_{j}$ to $B_{j}^i$. Thus by <Ref>, if we use the edges in $(B_j^i, B_{j+1}^i)$ and the edge $(v, B_{j+1}^i)$ we will exceed the valence of $a_{j}^i\in B_{j+1}^i$.  Therefore it is impossible to use the edges $(B_j^i, B_{j+1}^i)$ as well as the edge $(v_1, B_j^i)$ in an admissible matching.

* Now suppose we use the edges $(B_j^i, B_{j+1}^i)$ and the edge $(v_2,B_j^i)$ in an admissible matching.

* Assume $j<c_i-2$.  In this case, we can extend our local picture as follows, where $v_3$ and $v_4$ are additional vertices labeled $v$.

(vopt) at (0,0)$v_2$;
(bb) at (1,0)$B_j^i$;
(b11) at (1,-1)$B_{j+1}^i$;
(Bj2i) at (0,-1)$B_{j+2}^i$;
(v3) at (1,-2)$v_3$;
(a) at (0,-2)$B_{j+3}^i$;
(b) at (-1,-1)$v_4$;
[dashed] (vopt) to node[right]$Y_{I_{j+1}^i}$(b11);
(Bj2i) to node[left]$Y_{I_{j+2}^i}$ (vopt);
(vopt) to node[above]$I_j^i$(bb);
[double] (bb) to node[right]$A_j^i$(b11);
[double] (Bj2i) to node[above]$A_{j+1}^i$ (b11);
[dashed] (Bj2i) to node[right]$I_{j+2}^i$ (v3);
(v3) to node[below]$Y_{I_{j+3}^i}$ (a);
[double] (Bj2i) to node[right]$A_{j+2}^i$ (a);
(b11) to node[right]$Y_{I_{j+1}^i}$ (v3);
(b) to node[above]$Y_{I_{j+2}^i}$ (Bj2i);
[dashed] (a) to node[right]$I_{j+3}^i$ (b);

By the same argument as in part (a), we know $a^i_{j+1}\in B_{j+2}^i$. By <Ref>, $a^i_{j+1}$ must be covered in an admissible matching. We cannot use the edge $(v_2,B_{j+2}^i)$, as this would violate the valence of $v_2$ by <Ref>.  We also cannot use the edges $(B_{j+1}^i,B_{j+2}^i)$ as this would violate the valence of $a_{j}^i\in B_{j+1}^i$ by <Ref>.  If we use the edge $(v_4,B_{j+2}^i)$, we could not use the edge $(v_3,B_{j+3}^i)$ by Condition (<ref>) of Definition <ref>.  So, by <Ref>, we would have to use $(v_3,B_{j+1}^i)$.  This would violate the valence of $a^i_j\in B_{j+1}^i$, so we can't use $(v_4,B_{j+2}^i)$.  This means we must use $(B_{j+2}^i,B_{j+3}^i)$ to cover $a^i_{j+1}$.  However we are then unable to cover $v_3$, as using the edge $(v_3,B_{j+1}^i)$ would violate the valence of $a^i_j\in B_{j+1}^i$ and using the edge $(v_3,B_{j+3}^i)$ would violate the valence of $a^i_{j-2}\in B_{j+3}^i$.  Thus we  have a contradiction and cannot use the edges $(B_j^i, B_{j+1}^i)$ and the edge $(v_2,B_j^i)$ together in an admissible matching.

* Now assume $j=c_i-2$.  In this case we can extend  our local picture as follows.

(vopt) at (0,0)$v_2$;
(b11) at (1,-1)$B_{j+1}^i$;
(Bj2i) at (0,-1)$B_{j+2}^i$;
(v3) at (1,-2)$v_3$;
(a) at (0,-2)$a_i$;
[dashed] (vopt) to node[right]$I_{j+1}^i$(b11);
(Bj2i) to node[left]$Y_{I_{j+2}^i}$ (vopt);
[double] (Bj2i) to node[above]$A_{j+1}^i$ (b11);
[dashed] (Bj2i) to node[right]$I_{j+2}^i$ (v3);
(v3) to node[below]$Y_{I_{j+3}^i}$ (a);
[double] (Bj2i) to node[right]$A_{j+2}^i$ (a);
(b11) to node[right]$Y_{I_{j+1}^i}$ (v3);

The vertex $a_i=a_{c_i}^i=a_{j+2}^i$ is in the same configuration with respect to $B_{j+2}^i$ and $v_3$ as it was in the $j<c_i-2$ case. So the same arguments as in the $j<c_i-2$ case show we cannot use any of the edges adjacent to $B_{j+2}^i$ in an admissible matching at the same  time as the edges $(B_j^i, B_{j+1}^i)$ and the edge $(v_2,B_j^i)$.  Therefore, we can't use the edges $(B_j^i, B_{j+1}^i)$ and the edge $(v_2,B_j^i)$ together in an admissible matching in this case either.

* Finally, if $j=c_i-1$, we can extend  our local picture as follows.

(vopt) at (0,0)$v_2$;
(b11) at (1,-1)$B_{j+1}^i$;
(Bj2i) at (0,-1)$a_i$;
[dashed] (vopt) to node[right]$I_{j+1}^i$(b11);
(Bj2i) to node[left]$Y_{I_{j+2}^i}$ (vopt);
[double] (Bj2i) to node[above]$A_{j+1}^i$ (b11);

The vertex $a_i=a_{c_i}^i=a_{j+1}^i$ is in the same configuration with respect to $B_{j+1}^i$ and $v_2$ as it was in the $j<c_i-2$ case. So the same arguments as in the $j<c_i-2$ case show we cannot use any of the edges adjacent to $a_i$ in an admissible matching at the same time as the edges $(B_j^i, B_{j+1}^i)$ and the edge $(v_2,B_j^i)$.  Therefore, we can't use the edges $(B_j^i, B_{j+1}^i)$ and the edge $(v_2,B_j^i)$ together in an admissible matching in this final case.

Therefore, for an admissible matching $P$,
the only possibilities around $B_j^i$ are the following:

* $P$ uses $(v_1,B_j^i)$, and no other edges connected to $B_j^i$.

* $P$ uses $(v_2,B_j^i)$, and no other edges connected to $B_j^i$.

* $P$ uses all of $(B_j^i,B_{j+1}^i)$, and no other edges connected to $B_j^i$.

* $P$ uses all of $(B_j^i,B_{j-1}^i)$, and no other edges connected to $B_j^i$.

Now suppose $b_{i,c_i}^{\ell,k}\in B_{c_i}^i$ for $1\leq i\leq r$.  Then the relevant part of $\cal G_v$ will look as follows.

(b12) at (1,0)$B_{c_i}^i$;
(b22) at (2,0)$B_{c_i-1}^i$;
(v) at (1,1)$v_1$;
(vfix) at (2,-1)$v$;
(b11) at (1,-1)$a_i$;
[double]  (b12) to node[above]$A_{c_i-1}^i$ (b22);
(b12) to node[left]$Y_{I_{c_i}^i}$ (v);
[double] (b12) to node[left]$A_{c_i}^i$ (b11);
(vfix) to node[right]$Y_{I_{c_i-1}^i}$ (b22);
[dashed] (v) to node[right]$I_{c_i}^i$ (b22);
[dashed] (b12) to node[right]$I_{c_i}^i$ (vfix);
(vfix) to node[below] (b11);

The arguments from parts (a), (b), and (c) from the $1<j<c_i$ case show that for an admissible matching $P$,
the only possibilities around $B_{c_i}^i$ are the following:

* $P$ uses $(v^*,B_{c_i}^i)$, and no other edges connected to $B_{c_i}^i$.

* $P$ uses all of $(B_{c_i}^i,a_i)$, and no other edges connected to $B_{c_i}^i$.

* $P$ uses all of $(B_{c_i}^i,B_{c_i-1}^i)$, and no other edges connected to $B_{c_i}^i$.

Finally, suppose $b_{i,j}^{\ell,k}\in B_1^i$ for $1\leq i\leq r$.  Then the relevant part of $\cal G_v$ will look as follows, where the picture on the left is for $r=1$ and the picture on the right  is for $r=2$.

(vopt) at (0,0)$v_\opt$;
(v1) at (-2,1)$v_1$;
(b11) at (-1,1)$B_1^1$;
(b21) at (-1,0)$B_2^1$;
(ai) at (0,1)$\tilde{a}$;
(vopt) to node[above] (ai);
(ai) to (b11);
[double] (b21) to (b11);
(vopt) to node[right] (b21);
(v1) to node[right] (b11);
[double] (vopt) to node[right] (ai);
[dashed] (vopt) to node[right](b11);
[dashed] (v1) to node[right] (b21);

(vopt) at (0,0)$v_\opt$;
(b12) at (0,1)$B_1^2$;
(b22) at (0,2)$B_2^2$;
(v2) at (1,1)$v_2$;
(v1) at (-2,1)$v_1$;
(vfix) at (-1,2)$v_\fix$;
(b11) at (-1,1)$B_1^1$;
(b21) at (-1,0)$B_2^1$;
(ai) at (-0.25,0.75)$\tilde{a}$;
(vopt) to node[above] (b12);
[double]  (b12) – (b22);
(b12) to node[left] (v2);
(b12) to (b11);
(vfix) to node[right] (b22);
[double] (b21) to (b11);
(vopt) to node[right] (b21);
(v1) to node[right] (b11);
[dashed] (v2) to node[right] (b22);
[dashed] (b12) to node[right] (vfix);
(vfix) to node[below] (b11);
[double] (vopt) to node[right] (ai);
(-0.5,1) to node[right] (ai);
[dashed] (vopt) to node[right](b11);
[dashed] (v1) to node[right] (b21);

By Proposition <ref> of Definition <ref>, the edges in $(B_1^1,B_2^1)$ have to appear together in an admissible matching of $\cal G_v$.  By <Ref>, all vertices in $B_1^1$ have valence $1\oplus 0$.  This means that if $r=1$ an admissible matching of $\mathcal{G}_v$ must use either all of the $(B_1^1,B_2^1)$ edges or exactly one of the other edges adjacent to $B_1^1$.  Similarly, if $r=2$ then an admissible matching of $\mathcal{G}_v$ must use either all of the $(B_1^2,B_2^2)$ edges or exactly one of the other edges adjacent to $B_1^2$.

We have now seen that the admissible matchings of $\cal G_v$ are in bijection with perfect matchings of $\tilde{\cal G}_v$. We will next show that this bijection is weight preserving.  That is, we will show that each edge in $\tilde{\cal G}_v$ has the same weight as the corresponding set of edges in $\cal G_v$.

All boundary edges except for $(a_i,B^i_{c_i})$ correspond to either a hyperedge with weight $Y_{I^i_j}$, a set of edges with weight $1$ in $\cal G_v$, or a hyperedge with weight $Y_{I^i_j}$ and a set of edges with weight $1$ in $\cal G_v$.  Thus our claim is clear for these edges. If $r=2$, this is also clear for the edge $(B_1^1,B_1^2)$ because it  corresponds to an edge of weight $Y_{I_v}$ in $\cal G_v$.

For the edges $(a_i,B^i_{c_i})$, notice that after combining all component snake graphs, we get an hyperedge from $a_i$ to a subset of $B^i_{c_i}$ in $\cal G_v$ for each $I_w$ where $w\in \cal C_{a_i}$.  We also potentially get several edges of weight 1 from $a_i$ to vertices in $B^i_{c_i}\cap (\Gamma'_{>  a_i})^{\mathcal I}$.
Since $a_i=a^i_{c_1}$, the product of all these edges in $\cal G_v$ is $\prod_{w\in\cal C(a^i_{c_i})} Y_{I_w} = A_{c_i}^i$, which agrees with the weight of the edge $(a_i,B^i_{c_i})$ in $\tilde{\cal G}_v$.

We are now left to consider the edges $(B^i_j,B^i_{j+1})$ in $\tilde{\cal G}_v$.  The edge $(B^i_j,B^i_{j+1})$ corresponds to the edges $(b_{i,j}^{\ell,k},b_{i,j+1}^{\ell,k})$ from $\mathcal{G}_v$. Recall from <Ref>, in the component snake graphs, the edge $(b_{i,j}^{\ell,k},b_{i,j+1}^{\ell,k})$ can have three different possible weights.

* When $b_{i,j}^{\ell,k}\in N_{\Gamma'}(a^i_j)$, the edge $(b_{i,j}^{\ell,k},b_{i,j+1}^{\ell,k})$ has weight $1$.

* When $b^{\ell,k}_{i,j} \in N_{\Gamma'} (I_w) $
for some $w\in \mathcal{C}_{a^i_j} \setminus \{a^i_{j+1} \}$
, the edge $(b_{i,j}^{\ell,k},b_{i,j+1}^{\ell,k})$ is weighted by $Y_{I_w}$.

* When $b^{\ell,k}_{i,j}\in N_{\Gamma'}(I_{i,j+1})$, $(b_{i,j}^{\ell,k},b_{i,j+1}^{\ell,k})$ is weighted by $Y_{C_{i,j}^{\ell,k}}$ where $C_{i,j}^{\ell,k}$ is the connected component of $I^i_{j+1}\setminus [v,a^i_j]$ which contains a vertex adjacent to $b^{\ell,k}_{i,j}$.

In the second scenario, as we range over all possible $\ell\in A$, we get edges weighted by $Y_{I_w}$ for all $w\in \cal{C}_{a^i_j}\setminus{a^i_{j+1}}$ in $\cal G_v$. These edges will contribute to a weight of $\prod_{w\in \cal{C}_{a^i_j}\setminus{a^i_{j+1}}} Y_{I_w}$.

In the third scenario, since each $C_{i,j}^{\ell,k}$ is a connected component of $I^i_{j+1}\setminus [v,a^i_j]$ which contains a vertex adjacent to $b^{\ell,k}_{i,j}$, we know that $\bigcup_{\ell\in A} C_{i,j}^{\ell,k}$ is union of all the connected components of $I^i_{j+1}\setminus [v,a^i_j]$ which is adjacent to some vertex in $B^i_j = N_{\Gamma'}(I^i_{j})\setminus \{v\}$. This must be the entire set of $I^i_{j+1}\setminus [v,a^i_j]$, therefore the product of weights of these edges will be

\[\prod_{\ell\in A} Y^2_{C_{i,j}^{\ell,k}} = Y^2_{I^i_{j+1}\setminus [v,a^i_j]}.\]

Finally, when all the edges of the form $(b_{i,j}^{\ell,k},b_{i,j+1}^{\ell,k})$ are used together in $\cal G_{v}$, their weight will be

\[\left(\prod_{w\in \cal{C}_{a^i_j}\setminus{a^i_{j+1}}} Y_{I_w}\right)Y^2_{I^i_{j+1}\setminus [v,a^i_j]}\]

which is exactly $A^i_j$.

Therefore, the admissible matchings of $\cal G_v$ is in weight preserving bijection with perfect matchings of $\tilde{\cal G}_v$ whose edge weights are defined as in <Ref>.

Since the previous lemma compares our snake graph with a zigzag snake graph, we can use some a basic facts concerning the latter.  To state these, we first introduce some notation analogous to <cit.>. When considering a surface snake graph, we will say a boundary edge which is only incident to one diagonal and one tile is dominant and a boundary edge which is incident to one diagonal but two tiles is non-dominant. We provide an alternative definition for our snake graphs since the tile structure is not always clear.

Let $e$ be an edge of a snake graph $\mathcal{G}_v$ which only shares vertices with one diagonal. We say $e$ is dominant if $e$ is of the form $(v,w)$ where $w \in N_\Gamma(v)$.  Otherwise, we say $e$ is non-dominant.

Recall that when we form snake graphs for larger sets $S$ we glue along snake graph boundary edges of a snake graph of the form  $(v,w)$ where $v$ and $w$ are neighbors in $\Gamma$. These edges are always dominant edges.

The following lemma is a general statement for all zigzag snake graphs.

If $\mathcal{G}$ is a zigzag surface snake graph on $n$ tiles, then it has $n+1$ perfect matchings. In particular, it has $n-1$ perfect matchings such that each use exactly one internal edge and two perfect matchings that use complementary sets of boundary edges. Moreover, each of the boundary edge matchings uses exactly one dominant edge.

In particular, if $\mathcal{G}$ is a zigzag with tiles labeled by $\tau_{1},\ldots,\tau_{d}$ in order from southwest to northeast, with the internal edge between $\tau_{j}$ and $\tau_{j+1}$ labeled with $z_j$, the non-dominant edge on tile  $\tau_1$ (resp. tile $\tau_d$) labeled with $z_0$ (resp. $z_d$) and the dominant edge at on tile $\tau_1$ (resp. tile $\tau_d$) labeled with $u$ (resp. $v$), then
\[
\chi(\mathcal{G}) = \frac{u}{x_1} + \frac{v}{x_d} + \sum_{i=1}^{d-1} \frac{z_i}{x_i x_{i+1}}
\]

where $x_i$ is the cluster variable associated to $\tau_i$.

The fact that $\mathcal{G}$ has $n+1$ perfect matchings is well-known (see, for example, Section 4 of <cit.>). The perfect matchings of $\mathcal{G}$ are explicitly described in the proof of Theorem 3.4 of <cit.>, and the expansion for this particular zigzag snake graph follows from applying the definition of $\chi(\mathcal{G})$ given at the beginning of Section <ref>.

Let $\cal P_v$ denote the set of admissible matchings of $\cal G_v$. When $r=1$, we have

\[
\frac{1}{\ell(\cal{G}_v)}\sum_{P \in \cal{P}_v} \wt(P) = \frac{Y_{I_v}}{Y_{I^1_1}} + \sum_{j=1}^{c_1} \frac{A^1_j}{Y_{I^1_j}Y_{I^1_{j+1}}}
\]

In <Ref>, we showed that $\cal P_v$ is in bijection with the set of ordinary perfect matchings in $\tilde {\cal G_v}$. Hence, the conclusion follows from  <Ref> and <Ref>.

Let $\cal P_v$ denote the set of admissible matchings of $\cal G_v$. When $r=2$, we have

\[
\frac{1}{\ell(\cal{G}_v)}\sum_{P \in \cal{P}_v} \wt(P) = \frac{Y_{I_v}}{Y_{I^1_1}Y_{I^1_2}} + \sum_{j=1}^{c_1} \frac{A^1_j}{Y_{I^1_j}Y_{I^1_{j+1}}} + \sum_{j=1}^{c_2} \frac{A^2_j}{Y_{I^2_j}Y_{I^2_{j+1}}}
\]

In <Ref>, we showed that $\cal P_v$ is in bijection with the set of ordinary perfect matchings in $\tilde {\cal G_v}$. Hence, the conclusion follows from  <Ref> and <Ref>.

Comparing <Ref> to <Ref> shows that $\frac{1}{\ell(\mathcal{G}_v)} \sum_{P \in \mathcal{P}_v} \wt(P) = Y_v$ for any $v \in \Gamma$ with $r=1$. Comparing <Ref> to <Ref> shows that $\frac{1}{\ell(\mathcal{G}_v)} \sum_{P \in \mathcal{P}_v} \wt(P) = Y_v$ for any $v \in \Gamma$ with $r=2$.

§.§ r > 2

In this section we study the structure of admissible matchings on a singleton snake graph $\mathcal{G}_v$ such that $\vert N_{\Gamma}(v)\,\cap\,\Gamma^{\mathcal{I}}_{<v} \vert>2$. Unlike in Section <ref>, here we will have vertices with valence $a \oplus b$ where $a > 1$. We will first look at a family of intermediate graphs, which are the result of gluing some but not all component snake graphs. Let $\mathcal{H}_i$ denote the effect of gluing all component snake graphs $H_{\ell,k}$ where $a_1 \in [\ell,v]$ and $a_i \in [k,v]$, with $2 \leq i \leq r$. We call each $\mathcal{H}_i$ a branch snake graph. The center and right graphs in Figure <ref> are the two branch snake graphs for the snake graph $\mathcal{G}_3$ built in Example <ref>.

Recall that, for a vertex $v$ with $r \leq 2$, Lemma <ref> listed all possible valences of vertices of $\mathcal{G}_v$. When $r > 2$, this lemma is true for all vertices except $v_\fix$, which has valence $(r - 1) \oplus 0$. From this observation, we have that Proposition <ref> is also true for $r > 2$.

Let  $P$ be an admissible matching of $\mathcal{G}_v$ where $r > 2$. Let $e_1$ and $e_2$ be two internal edges which both share vertices with diagonals $d_1$ and $d_2$. Then, $e_1$ and $e_2$ are either both in $P$ or neither are in $P$.

The only difference between this setting and a snake graph $\mathcal{G}_v$ where $r \leq 2$ is that there is a vertex, $v_\fix$, with valence $(r -1) \oplus 0$. There are no internal edges incident to $v_\fix$ and the same arguments as for the $r \leq 2$ will hold here.

Along the lines of  <Ref>, we will show that admissible matchings of $\mathcal{H}_i$ are in weight-preserving bijection with the set of perfect matchings of a snake graph.

Let $\tilde{\mathcal{G}}_v^i$ be the snake graph of the same shape as that in Figure <ref> and with weights given by replacing all superscript “2"s with “$i"s$ and all “$c_2$"s with “$c_i$"s. The   set of admissible matchings of $\mathcal{H}_i$ is in weight-preserving bijection with the set of perfect matchings of $\tilde{\mathcal{G}}_{v}^i$.

For vertices other than $v_\fix$ and $v_\opt$, the same logic as Lemma <ref> holds for considering the valence of vertices in $\mathcal{H}_i$. The vertices $v_\fix$ and $v_\opt$ will have valence $1 \oplus 0$ in $\mathcal{H}_i$ since each is adjacent to the same edges in all the component snake graphs $H_{\ell,k}$ which glue together to form $\mathcal{H}_i$.
Then, we can apply similar reasoning as in the proof of Lemma <ref> for the $r = 2$ case.

We will now compare admissible matchings of the snake graph $\mathcal{G}_v$ to matchings of the branch snake graphs $\mathcal{H}_i$. It will not always be true that an admissible matching of $\mathcal{G}_v$ will decompose to admissible matchings of each $\mathcal{H}_i$. For example, in Figure <ref>, we give an admissible matching of the snake graph $\mathcal{G}_3$ from Example <ref> and its restrictions to its two branch snake graphs, $\mathcal{H}_2$ and $\mathcal{H}_3$. We notice that the restriction to $\mathcal{H}_3$ is an admissible matching of $\mathcal{H}_3$ but the restriction to $\mathcal{H}_2$ is not a matching.

[style=label] (0) at (-6.75, 5) $3$;
[style=label] (1) at (-2, 5) $4$;
[style=label] (2) at (-6.75, 0) $8'$;
[style=label] (3) at (-2, 1) $9$;
[style=label] (4) at (-2.75, 0) $7'$;
[style=label] (5) at (-3.5, -1) $0'$;
[style=label] (6) at (-1.25, 2) $1'$;
[style=label] (7) at (-1.25, 6) $1$;
[style=label] (8) at (2.5, 6) $2$;
[style=label] (9) at (2.5, 2) $5$;
[style=label] (10) at (-6.75, -4.25) $8$;
[style=label] (11) at (-1.25, -4.25) $3$;
[style=none] (12) at (0.25, 4.5) ;
[style=label-s] (13) at (-4.5, 2.5) $I_4$;
[style=label-s] (14) at (-2.75, 2) $I_7$;
[style=none] (15) at (-2, -2) ;
[style=label-s] (16) at (-5.5, 0) $I_3$;
[style=label-s] (18) at (-4, -2) $I_8$;
[style=none] (21) at (-1.75, -2.75) ;
[style=label] (26) at (0.25, -4.25) $=v_{{opt}}$;
[style=label-s] (27) at (-3, 4.5) $1\oplus 1$;
[style=label-s] (28) at (-4.25, 3.5) $I_1$;
[style=label-s] (29) at (1, 4.5) $I_2$;
[style=label-s] (30) at (-2, -5) $1\oplus 1$;
[style=label-s] (31) at (-7.5, 5.5) $2\oplus 0$;
[style=label] (32) at (5.5, 4.25) $3$;
[style=label] (34) at (5.5, 0.5) $8'$;
[style=label] (38) at (9.25, 0.5) $1'$;
[style=label] (39) at (9.25, 4.25) $1$;
[style=label] (40) at (12.75, 4.25) $2$;
[style=label] (41) at (12.75, 0.5) $5$;
[style=label] (42) at (5.5, -3.25) $8$;
[style=label] (43) at (9.25, -3.25) $3$;
[style=label-s] (48) at (7, 0.5) $I_3$;
[style=label-s] (49) at (7.5, -1.5) $I_8$;
[style=label-s] (57) at (7.5, 2.75) $I_1$;
[style=label-s] (58) at (11.5, 2.75) $I_2$;
[style=label] (61) at (17, 4.75) $3$;
[style=label] (62) at (22.25, 4.75) $4$;
[style=label] (63) at (17, 0.5) $8'$;
[style=label] (64) at (22.25, 1.5) $9$;
[style=label] (65) at (21.5, 0.5) $7'$;
[style=label] (66) at (20.75, -0.5) $0'$;
[style=label] (71) at (17, -3.75) $8$;
[style=label] (72) at (22.25, -3.75) $3$;
[style=label-s] (74) at (19.5, 2.75) $I_4$;
[style=none] (75) at (21.5, 2.25) ;
[style=none] (76) at (22.25, -2) ;
[style=label-s] (77) at (18.5, 0.5) $I_3$;
[style=label-s] (78) at (19.75, -1.75) $I_8$;
[style=none] (79) at (22.25, -2.5) ;
[style=label-s] (85) at (22, 5.25) $1\oplus 1$;
[style=label-s] (89) at (17, 5.25) $2\oplus 0$;

(0) to (1);
[style=m1] (1) to (3);
[style=m1] (0) to (2);
(7) to (8);
[style=m1] (8) to (9);
[style=e1] (9) to (6);
(6) to (7);
[style=m1] (0) to (7);
(2) to (10);
[style=m1] (10) to (11);
[style=m1] (11) to (6);
[style=e3] (2) to (16);
[style=e3, in=-150, out=0, looseness=0.75] (16) to (3);
[style=e3, in=-180, out=0] (16) to (4);
[style=e3, in=150, out=0, looseness=0.50] (16) to (5);
[style=e3, in=210, out=0, looseness=0.75] (16) to (6);
[style=e4, in=-75, out=105] (15.center) to (4);
[style=e4, in=-30, out=105] (15.center) to (5);
[style=m1] (14) to (4);
[style=m1] (14) to (5);
[style=d4] (0) to (13);
[style=d4, in=-180, out=-45, looseness=0.75] (13) to (3);
[style=d4] (13) to (4);
[style=d4, in=90, out=-45, looseness=0.75] (13) to (5);
[style=d8] (2) to (18);
[style=d8] (18) to (11);
[style=d2, in=0, out=135, looseness=0.75] (12.center) to (1);
[style=e4, in=270, out=105] (15.center) to (3);
(21.center) to (15.center);
[style=e4] (21.center) to (11);
(21.center) to (15.center);
[style=d1] (0) to (6);
[style=m1] (14) to (1);
[style=d2] (7) to (12.center);
[style=d2] (12.center) to (9);
[style=m1] (32) to (34);
(39) to (40);
[style=m1] (40) to (41);
[style=e1] (41) to (38);
(38) to (39);
[style=m1] (32) to (39);
(34) to (42);
[style=m1] (42) to (43);
[style=m1] (43) to (38);
[style=e3] (34) to (48);
[style=e3] (48) to (38);
[style=d8] (34) to (49);
[style=d8] (49) to (43);
[style=d1] (32) to (38);
(61) to (62);
[style=m1] (62) to (64);
[style=m1] (61) to (63);
(63) to (71);
[style=m1] (71) to (72);
[style=e3] (63) to (77);
[style=e3, in=-150, out=0, looseness=0.75] (77) to (64);
[style=e3, in=-180, out=0] (77) to (65);
[style=e3, in=150, out=0, looseness=0.75] (77) to (66);
[style=e4, in=-60, out=105] (76.center) to (65);
[style=e4, in=-30, out=105] (76.center) to (66);
[style=m1] (75.center) to (65);
[style=m1] (75.center) to (66);
[style=d4] (61) to (74);
[style=d4, in=-180, out=-45, looseness=0.75] (74) to (64);
[style=d4] (74) to (65);
[style=d4, in=90, out=-45, looseness=0.75] (74) to (66);
[style=d8] (63) to (78);
[style=d8] (78) to (72);
[style=e4] (76.center) to (64);
(79.center) to (76.center);
[style=e4] (79.center) to (72);
(79.center) to (76.center);
[style=m1] (75.center) to (62);
[style=d2] (39) to (41);

From left-to-right, we have an admissible matching $P$ of the snake graph $\mathcal{G}_3$ from Example <ref>, the restriction of $P$ to $\mathcal{H}_2$, and the restriction of $P$ to $\mathcal{H}_3$, where we order the neighbors of $3$ as $a_1 = 8, a_2 = 2$ and  $a_3 = 4$.

The reason this fails to be a matching comes from the fact that vertices $v_\fix$ and $v_\opt$ have valence $1 \oplus 0$ in the branch snake graphs. When these restrictions are not matchings, they will have a common form, which we will call the distinguished non-matching of $\mathcal{H}_i$.  This set of edges will include the edges weighted $Y_{I_1^i}$ and $Y_{I_2^1}$ incident to the vertex $v$ which lies on diagonal $I_1^1$ and similarly the edges $Y_{I_1^1}$ and $Y_{I_2^i}$ incident to the vertex $v$ on diagonal $I_1^i$, as described above.  Then, for all $2 \leq j \leq c_1$, it will include the boundary edge weighted $Y_{I_{j+1}^1}$ incident to the vertex $v$ on diagonal $I_j^1$ and for $2 \leq j \leq c_i$, it will include the boundary edge weighted $Y_{I_{j+1}^i}$ incident to the vertex $v$ on diagonal $I_j^i$. Finally, this matching includes both dominant edges. The set of edges highlighted in $\mathcal{H}_2$ in Figure <ref> is an example of a distinguished non-matching.  The weight of this matching will be $Y_{I_1^1} Y_{I_2^1} \cdots Y_{I_{c_1}^1} Y_{I_1^i} Y_{I_2^i} \cdots Y_{I_{m_i+1}^i}$ so that its weight is equal to $\ell(\mathcal{H}_i)$.

In the following, let $\mathcal{P}_v$ denote the set of admissible matchings of $\mathcal{G}_v$. Let $\mathcal{P}_v^1$ denote the set of admissible matchings of $\mathcal{G}_v$ which restrict to an admissible matching for each $\mathcal{H}_i$. Let $\mathcal{P}_v^i$, $2 \leq i \leq r$, denote the set of admissible matchings which only restrict to an admissible matching of $\mathcal{H}_i$ and restrict to the distinguished non-matching for all $\mathcal{H}_j, j \neq i$. Since $r > 2$, it follows from the definitions that  $\mathcal{P}_v^i$ and $\mathcal{P}_v^j$ are disjoint for $i \neq j$.

The sets $\mathcal{P}_v^1,\ldots,\mathcal{P}_v^r$ partition $\mathcal{P}_v$.

We have noted that the vertices $v_\opt$ and $v_\fix$ are the only vertices which have different valences in $\mathcal{G}_v$ and $\mathcal{H}_i$. In $\mathcal{H}_i$, the vertex $v_{\fix}$ is incident to boundary edges weighted $Y_{I_1^1}$ and $Y_{I_2^i}$, the vertex $v_\opt$ is incident to boundary edges weighted $Y_{I_2^1}$ and $Y_{I_1^i}$, and both vertices have valence $1 \oplus 0$. Now, we turn to $\mathcal{G}_v$.
The vertex $v_\fix$ is incident to $r$ edges, with weights $Y_{I_1^1}, Y_{I_2^2}, Y_{I_2^3},\ldots, Y_{I_2^r}$ and has valence $(r-1) \oplus 0$. The vertex $v_\opt$ has valence $(1 \oplus \deg_{\Gamma'}(v)-2)$ and is incident to $\deg_{\Gamma'}(v)$ edges, where $r$ have weights $Y_{I_2^1},Y_{I_1^2},Y_{I_1^3},\ldots,Y_{I_1^r}$ and the remaining $\deg_{\Gamma'}(v)-r$ are of the form $(v_\opt,a_i)$ for $r+1 \leq i \leq \deg_{\Gamma'}(v)$.

Note that every admissible matching of $\mathcal{G}_v$ must include all but one edge incident to $v_\fix$. We will show that $\mathcal{P}_v^1$ is the set of admissible matchings which does not contain the edge $Y_{I_1^1}$ incident to $v_\fix$ and $\mathcal{P}_v^i$ is the set of admissible matchings which does not use the edge $Y_{I_2^i}$ incident to $v_\fix$.

Suppose first that a matching $P$ does not contain the edge $Y_{I_1^1}$ incident to $v_\fix$. Then, it must contain all edges in the set $\{Y_{I_2^i}| 2 \leq i \leq r\}$ which are incident to $v_\fix$. We claim any such matching will restrict to an admissible matching on each $\mathcal{H}_i$. We can show this by showing that we will not use both the edge weighted $Y_{I_2^1}$ incident to $v_\opt$ and any edge $Y_{I_1^i}$ for $ 2 \leq i \leq r$ since this is the only other vertex whose valence is different in $\mathcal{G}_v$ and $\mathcal{H}_i$.

Suppose for sake of contradiction that the edges from $P$ incident to $v_\opt$ consists of the edge $Y_{I_2^1}$  and some nonempty subset of the edges $\{Y_{I_1^i}\vert 2 \leq i \leq r\}$. Once the set contains at least one edge $Y_{I_1^i}$ incident to $v_\opt$, we know it cannot contain the edge $Y_{I_v}$ because all vertices incident to $I_v$ have valence $1 \oplus 0$.  Therefore, the matching must include all edges $(v_\opt,a_i)$ for $r+1 \leq i \leq \deg_{\Gamma'}(v)$. This means that there must also be at least one value $1 \leq i \leq r$ such that the edge $Y_{I_1^i}$ incident to $v_\opt$ is not in $P$.

Suppose without loss of generality that this edge is $Y_{I_1^2}$. As in Section <ref>, let $B_j^i = N_{\Gamma'}(I_j^i) \backslash \{v\} = \{b_{i,j}^{\ell,k} : \ell \in A, k \in B\}$ for $1 \leq j \leq c_1$.   Now, we claim that there is no way to find an edge in $P$ which is incident to any vertex in $B_1^2$. The local picture is as follows.

(vopt) at (0,0)$v_\opt$;
(b12) at (1,0)$B_1^2$;
(b22) at (2,0)$B_2^2$;
(v) at (1,1)$v$;
(vfix) at (2,-1)$v_\fix$;
(b11) at (1,-1)$B_1^1$;
(vopt) to node[above]$Y_{I_1^2}$ (b12);
(b12) – (b22);
(b12) to node[left]$Y_{I_1^2}$ (v);
(b12) to node[right]$Y_{I_v}$ (b11);
(vfix) to node[right]$Y_{I_2^2}$ (b22);
[dashed] (v) to node[right]$I_2^2$ (b22);
[dashed] (b12) to node[right]$I_1^2$ (vfix);
(vfix) to node[below]$Y_{I_1^1}$ (b11);
[dashed] (vopt) to node[right]$I_1^1$(b11);

We already assume we cannot use the edge between $v_\opt$ and $B_1^2$, and we have shown we cannot use the edge $Y_{I_v}$. We claim we cannot use any of the edges between the sets $B_2^2$ and $B_1^2$. Note that we already use the edge between $v_\fix$ and $B_2^2$, and if we included one edge between $B_2^2$ and $B_1^2$, by Lemma <ref> we would need to include all such edges. We can immediately see that this configuration would violate the valence of at least one vertex in $B_2^2$ unless $B_2^2 = \{a_1^2\}$ and the valence of $a_1^2$ is $1 \oplus a$ for $a > 0$. However, from the proof of Lemma <ref> we know that $a_1^2$ is incident to $a+1$ distinct edges of the form $(a_1^2, B_1^2)$, so that including all of these edges plus the edge $(v_\fix, B_2^2)$ would again violate the valence of $a_1^2$.

Finally, we cannot use the edge incident to $B_1^2$ which has weight $Y_{I_1^2}$ and is incident to a vertex $v \neq v_\opt$ because, since we use the edge $Y_{I_2^2}$ incident to $v_\fix$, this would violate Condition (<ref>) in Definition <ref>. In conclusion, when we know that $P$ contains the edges incident to $v_\fix$ labeled $I_2^2,\ldots,I_r^2$, then $P$ will restrict to an admissible matching on all $\mathcal{H}_i$.

Next, suppose that the edge with weight $Y_{I_1^1}$ incident to $v_\fix$ is in the matching $P$. Then one edge $Y_{I^i_2}$ must not be in the matching; suppose without loss of generality this is $I_2^2$. Now, we claim that $P$ restricts to the distinguished non-matching on each $\mathcal{H}_i$ for $i \geq 3$ and $P$ restricts to an admissible matching of $\mathcal{H}_2$. This will be equivalent to showing that $P$ contains all edges incident to $v_\opt$ except the edge $Y_{I_1^2}$. In the following, we assume $c_1 \geq 3$; the claim can be checked directly for small values of $c_1$.

Suppose for sake of contradiction that one of these edges is not in $P$. Since the edge $Y_{I_1^1}$ incident to $v_\fix$ is in $P$, we know that we cannot use the edge $Y_{I_v}$. Thus, we know immediately that we must use all edges $(v_\opt,a_i)$ for $r+1 \leq i \leq \deg_{\Gamma'}(v)$.

If an edge $Y_{I_1^i}$, $i > 2$, incident to $v_\opt$ is not in $P$, then with a very similar argument to the previous case where edges $Y_{I_2^2},\ldots,Y_{I_2^r}$ incident to $v_\fix$ were in $P$, we can show that there is no way to find an edge in $P$ which is incident to any vertex in $ B_1^3$. So all edges $Y_{I_1^i}$, $i > 2$ must be in $P$.

Finally, suppose that the edge $Y_{I_2^1}$ incident to $v_\opt$ is not in $P$. We claim that there cannot be an edge in $P$ which is incident to $B_2^1$. The local picture is as follows.

(vopt) at (0,0)$v$;
(b12) at (1,0)$B_2^1$;
(b22) at (2,0)$B_1^1$;
(v) at (1,1)$v_\opt$;
(vfix) at (2,-1)$v$;
(b11) at (1,-1)$B_3^1$;
(vreallyfix) at (3,0)$v_\fix$;
(vopt) to node[above]$Y_{I_2^1}$ (b12);
(b12) – (b22);
(b12) to node[left]$Y_{I_2^1}$ (v);
(b12) to (b11);
(vfix) to node[right]$Y_{I_1^1}$ (b22);
[dashed] (v) to node[right]$I_1^1$ (b22);
[dashed] (b12) to node[right]$I_2^1$ (vfix);
(vfix) to node[below]$Y_{I_3^1}$ (b11);
[dashed] (vopt) to node[right]$I_3^1$(b11);
(b22) to node[above]$Y_{I_1^1}$ (vreallyfix);

First, note that since the edge $Y_{I_1^1}$ incident to $v_\fix$ is in $P$, we must include the edge $Y_{I_1^3}$ incident to the vertex $v$ on diagonal $I_2^1$. The other edge incident to this vertex would create a valence violation at $B_1^1$ as all vertices in $B_1^1$ have valence  $1 \oplus 0$. Therefore, $P$ cannot contain the edge $Y_{I_2^1}$ which is incident to $B_2^1$ and the vertex $v$ on diagonal $I_3^1$ by Condition (<ref>) of Definition <ref>.  We cannot use any edge between $B_2^1$ and $B_1^1$ because this would also create a valence violation for vertices in $B_1^1$. We also claim we cannot use any edges  between $B_3^1$ and $B_2^1$. As in the previous case, by Lemma <ref>, we would need to include all internal edges which share vertices with diagonals $I_3^1$ and $I_2^1$ and this would still create a valence violation since we also have included  the edge $Y_{I_3^1}$. Therefore, we have shown that when an admissible matching does not contain $Y_{I_1^i}$,then it will restrict to an admissible matching of $\mathcal{H}_i$ and the distinguished non-matching of $\mathcal{H}_j, j \neq i$.

We include the following corollary to Lemma <ref> which will be useful in Section <ref>.
Recall from Proposition <ref> that for $a_i \in N_{\Gamma'}(v)$, a unique boundary edge $(v,a_i)$  is guaranteed to exist in $\mathcal{G}_v$.

For each $a_i \in N_{\Gamma'}(v)$, there is a unique admissible matching of $\mathcal{G}_v$ which does not include the boundary edge $(v,a_i)$.

This is immediate for the case where $r=1$  and $r=2$ by Lemma <ref>. Now, suppose $r > 2$. We first consider the case $1<i \leq r$. Then, a matching which does not include $(v,a_i)$ could only be in the sets $\mathcal{P}_v^1$ and $\mathcal{P}_v^i$. However, in the proof of Lemma <ref>, we show that all matchings in $\mathcal{P}_v^1$ include the edge $(v,a_i)$. Combining Lemmas <ref> and <ref> shows that there is exactly one matching in $\mathcal{P}_v^i$ which includes the non-dominant boundary edges incident to diagonal $I_{c_i}^i$ instead and therefore does not include edge $(v,a_i)$.

If $i = 1$, then we can see that every matching in $\mathcal{P}_v^i$ for $i > 1$ contains this boundary edge. Thus, we will similarly get exactly one matching not containing $(v,a_1)$ in $\mathcal{P}_v^1$.

Now, if $r<i\leq p$ then in $\mathcal{G}_v$ this vertex is incident to edge $(a_i,v)$ and another edge with weight $Y_{I_v}$. We also know that this vertex has valence $1 \oplus 0$ so exactly one of these edges will be present in any admissible matching. From the proof of Lemma <ref>, we see that there is a unique matching using the edge $Y_{I_v}$, so we are done.

The proof of Theorem <ref> in the case of $S = \{v\}$ follows from Lemma <ref>.

By Lemma <ref>, we can separate the sum in the definition of $\chi(\mathcal{G}_v)$ according to how the various admissible matchings of $\mathcal{G}_v$ restrict on the branch snake graphs. It remains to evaluate the weights of these sums.

We first evaluate the possible weights of the matchings in $\mathcal{P}_v^1$. Repeated use of Condition (<ref>) in Definition <ref> shows that, given a vertex $v$ incident to diagonal $I_j^i$ for $i>1$, the matching can only contain the boundary edge at this vertex with weight $Y_{I_{j+1}^i}$. This includes the dominant edge incident to each diagonal $I_{c_i}^i$.

None of the edges specified so far are incident to any diagonal $I_j^1$. There are in fact no restrictions on which edges in this subgraph can or cannot be included in the matching. Therefore, we can apply the reasoning of Lemma <ref> to list all possible ways the matching can look for this section. Since the contributions for the edges outside this subgraph  cancel with the part of the denominator coming from diagonals outside this subgraph, the sum of the weights of such matchings again follows from Lemmas <ref> and <ref>. Thus, we have

\[
\frac{1}{\ell(\mathcal{G}_v)}\sum_{P \in \mathcal{P}_v^1} wt(P) = \frac{Y_{I_v}}{Y_{I_1^1}\cdots Y_{I_1^r}} + \sum_{j=1}^{c_1} \frac{A_j^1}{Y_{I_j^1}Y_{I_{j+1}^1}}.
\]

Now, consider $\mathcal{P}_v^i$ for $2 \leq i \leq r$. Since any matching in $\mathcal{P}_v^i$ restricts to the distinguished non-matching on $\mathcal{H}_j$ for $j \geq 2, j \neq i$, we can simply consider the restriction to $\mathcal{H}_i$. In Lemma <ref>, we have shown that any $P \in \mathcal{P}_v^i$ contains the edge $Y_{I_1^1}$ at $v_\fix$ and the edge $Y_{I_2^1}$ at $v_\opt$. Using Condition (<ref>) of Definition <ref> as we have previously, we can see that we will have edges $Y_{I_3^1},\ldots,Y_{I_{c_1}^1}$ and the dominant edge incident to diagonal $I_{c_1}^1$. No other edges are forced, so we see that it is possible to find a $P$ which restricts to any admissible matching of $\mathcal{H}_i$ which includes the set of internal edges incident to diagonals $I_j^i$ and $I_{j+1}^i$ for $1 \leq j < c_i$ or the admissible matching which includes the set of non-dominant edges incident to diagonal $I_{c_i}^i$. By combining Lemmas <ref> and <ref>, these are all the possibilities. Thus, we have,

\[
\frac{1}{\ell(\mathcal{G}_v)}\sum_{P \in \mathcal{P}_v^i} wt(P) = \sum_{j=1}^{c_i} \frac{A_j^i}{Y_{I_j^i}Y_{I_{j+1}^i}}.
\]

Thus, from Corollary <ref>, we can conclude
\[
\frac{1}{\ell(\mathcal{G}_v)} \sum_{P \in \mathcal{P}_v} wt(P) = \frac{1}{\ell(\mathcal{G}_v)} \sum_{P \in \mathcal{P}_v^1} wt(P) + \sum_{i=2}^r \sum_{P \in \mathcal{P}_v^i} wt(P) = Y_v.
\]

§ PROOF FOR GENERAL WEAKLY ROOTED SETS

Now, we turn to proving Theorem <ref> for general weakly rooted sets. The main proof will be contained in Section <ref> where we will induct on $\vert \bar{S} \vert$.  We complete the base case $\vert \bar{S} \vert = 1$ in Section <ref>, building on the machinery we developed for when $\vert S \vert = 1$ in Section <ref>.

§.§ Proof for Sets with Singleton Weakly Rooted Portions

We provide here a proof of Theorem <ref> for a weakly rooted set $S = J_{\mathbf{m}}$. We constructed snake graphs for two such sets in Example <ref> as intermediate steps.

Recall from the statement of Proposition <ref>, for a vertex $v$ and a tuple $\mathbf{m} = (m_1,\ldots,m_r)$, that we set $J_{\mathbf{m}} = \{v\} \cup I_{m_1}^1(v) \cup \cdots \cup I_{m_r}^r(v)$. It is clear that $J_{\mathbf{m}}$ is weakly rooted and  $\overline{J_{\mathbf{m}}} = v$. Moreover, every set $S$ with $\overline{S} = \{v\}$ will be of the form $S = J_{\mathbf{m}}$.

First, we show that Proposition <ref> remains true in this setting.

Let $S = J_{\mathbf{m}}$ be a weakly rooted set with $\bar{S} = \{v\}$ and let $P$ be an admissible matching of $\mathcal{G}_S$. Let $e_1$ and $e_2$ be two internal edges which both share vertices with diagonals $d_1$ and $d_2$. Then, $e_1$ and $e_2$ are either both in $P$ or neither are in $P$.

We can repeat the logic as in the proof of Proposition <ref> but we must take a bit more care in the first case for any $i$ such that $m_i \leq c_i$.  In order for edges between $B_{j}^i$ and $B_{j-1}^i$ to be internal, we must have $j<m_i$.  Therefore, let $m_i > 2$ and suppose we take a proper subset of the internal edges which share vertices with diagonals $I_{m_i-1}^i$ and $I_{m_i-2}^i$. A consequence of Condition (<ref>) and Lemma <ref> is that every vertex in $B_{m_i-1}^i$ is either not covered or is covered a maximal number of times. Since this is a proper subset, there must be vertices in $B_{m_i-1}^i$ which have not been covered at all.   We cannot cover these with the boundary edge $Y_{I_{m_i-1}^i}$ since this would violate the valence of the vertices which are already incident to these internal edges. If $m_i\neq c_i+1$, then the other set of boundary edges incident to $B_{m_i-1}^i$ consists of a set of edges incident to $a_{m_i-1}^i$ and other edges disjoint from these. This is because these edges were the internal edges which shared vertices with diagonals $I_{m_i-1}^i$ and $I_{m_i}^i$ in $\mathcal{G}_v$.

It is possible that we could cover the necessary vertices in $B_{m_i-1}^i$ using some of the edges between $B_{m_i}^i$ and $B_{m_i-1}^i$. However, in order to not violate the valences of any vertex in $B_{m_i-1}^i$, this would require a proper subset of these edges. The only other edge, $Y_{I_{m_i}^i}$, incident to the set of vertices $B_{m_i}^i$ is incident to all of these vertices. Condition (<ref>) of Definition <ref> forces us to include either all or none the edges incident to $a_{m_i-1}^i$ in an admissible matching. By the same logic as in Lemma <ref>, an admissible matching that contains all of these edges cannot also contain any other edge incident to $a_{m_i-1}^i$. Because all vertices in $B_{m_i}^i$ other than $a_{m_i-1}^i$ have valence $1\oplus 0$, using edge $Y_{I_{m_i}^i}$ would violate the valence of at least one vertex in $B_{m_i}^i$. Therefore, we cannot include a proper subset of the internal edges which share vertices with diagonals $I_{m_i-1}^i$ and $I_{m_i-2}^i$.

As in the proof of Proposition <ref>, if we include proper subsets of the set of internal edges which share vertices with $I_{m_i-j}^i$ and $I_{m_i-j-1}^i$ with $1<j<m_i+1$, we would be forced back to this case.

Let $S$ be a weakly rooted set such that $\bar{S} = \{v\}$.  Then  $S= J_{\mathbf{m}}$ for some ${\bf m}=(m_1,\dots,m_r)$. In Proposition <ref>, we have an exact formula for $Y_S$, so all that remains is to show that this formula agrees with the weighted sum indexed by all admissible matchings of $\mathcal{G}_S$.

We reorder the neighbors of $v$ as $a_1,\ldots,a_r$ so that for some $0 \leq t \leq r$, we have $m_i > 1$ for $1 \leq i \leq t$ and $m_i = 1$ for $t+1 \leq i \leq r$.

First, note that if $t = r$, then we can use the same reasoning as in Sections <ref> and <ref> to understand the admissible matchings of $\mathcal{G}_S$. The only difference is that, when $m_i < c_i+1$, instead of a dominant edge $(v,a_i)$ we have an edge with weight $Y_{I_{m_i}^i}$ that shares vertices with diagonal $I_{m_i-1}^i$. For a given $i$, every admissible matching will include this edge except the boundary edge with weight $Y_{I_{m_i-2}^i}$ which shares vertices with the diagonal $I_{m_i-1}^i$. Such a matching must also include the edges whose product of weights is $A_{m_i-1}^i$, and no other matching will use these edges. Comparing these observations with the formula given in Proposition <ref> shows the claim in this case.

The other extreme value of $t$ is when $t = 0$. In this case, $S = I_v$ and $\mathcal{G}_S$ is the single hyperedge labeled $I_v$ where all vertices have valence $1 \oplus 0$. The claim is immediate.