# Complexity of Chess Domination Problems

Alexis Langlois-Rémillard  Department of applied mathematics, computer science
and statistics, Faculty of Sciences, Ghent University, Ghent, Belgium
and ScaDS.IA Leipzig, Universität Leipzig, Leipzig, Germany  alexis.langlois-
<EMAIL_ADDRESS>,  Christoph Müßig  Technische Universität München,
Garching b. München, Germany<EMAIL_ADDRESS>and  Érika Roldán-Roa  Max
Planck Institute for Mathematics in the Sciences, Inselstraße 22, Leipzig,
Germany<EMAIL_ADDRESS>
###### Abstract.

We study different domination problems of attacking and non-attacking rooks
and queens on polyominoes and polycubes of all dimensions. Our main result
proves that maximal domination is NP-complete for non-attacking queens and for
non-attacking rooks on polycubes of dimension three and higher. We also
analyse these problems for polyominoes and convex polyominoes, conjecture the
complexity classes and provide a computer tool for investigation. We have also
computed new values for classical queen domination problems on chessboards
(square polyominoes). For our computations, we have translated the problem
into an integer linear programming instance. Finally, using this computational
implementation and the game engine Godot, we have developed a video game of
minimal domination of queens and rooks on randomly generated polyominoes.

03D15, 05B40, 05B50, 00A08, 68Q17, 68R05, 68R07

###### Key words and phrases:

Keywords: Art Gallery Theorem, NP-Completion, NP-Hardness, Polyomino,
Computational Geometry, Visibility Coverage, Guard Number, Domination Problem,
N-Queens Problem, Linear Programming

###### 1991 Mathematics Subject Classification:

MSC 2020:

##  1\. Introduction

One of the most ancient and famous enumeration problems involving a chessboard
and chess pieces is the 8-queens problem that was first stated by Max Bezzel
in 1848—see [3, 6, 16] for accounts of its history. This problem asks to find
the number of different ways of placing 8 queens on a chessboard without
attacking each other (queens can attack vertically, horizontally, and in
diagonals). Note that $8$ is the maximal number of non-attacking queens that
can be placed on an $8\times 8$ chessboard. Many problems stemmed from this
puzzle, some of them are: the $n$–queens problem [1], that is, the 8–queens
problem generalized to $n\times n$ chessboards for $n\geq 1$; the completion
problem [9, 10, 23], which asks if it is possible to complete a given set of
non-attacking queens on a $n\times n$ chessboard to a non-attacking set with
$n$ queens; and the minimal domination problem [14, Appendix], which consists
in finding the minimal number of queens necessary to guard or dominate a
chessboard—the known values of this sequence can be found in [26, A075324].

In this paper, we study domination problems on polyominoes and polycubes111In
1954, Solomon W. Golomb defined a polyomino as a _finite rook-connected subset
of squares of the infinite checkerboard_ [11]. A $d$–polycube is its extension
to dimension $d$. by rooks and queens. This problem is also known as the art
gallery problem on polyominoes [2, 4, 25]. In recent work, the NP-hardness of
the minimal domination by rooks and queens was proven for $d$–polycubes for
$d\geq 2$ [2]—see Table 3. In the same paper, the authors studied the maximal
non-attacking rooks set problem on polyominoes, and they proved that it is in
P.

We note that the queen problem complexity has been considered on walled
chessboards, equivalent to edge-polyomino [21].

Our first main results will be to prove the NP-completeness of the minimal
domination of attacking, and non-attacking, rooks and queens on polycubes,
extending [2, Thms 3,4]. To give a perspective, let us note that the minimal
domination problem of rooks on a square chessboard is trivially polynomial: a
rook is needed in each row and each column so filling the diagonal solves it
for any chessboard; and the minimal domination problem for queens on the
chessboard has been studied for the last 150 years, and yet we still do not
know whether there exists a polynomial-time algorithm to find the minimal
number of queens needed to dominate a chessboard.

We also study the problem of finding maximal independent sets of queens or
rooks on polyominoes, that is, the maximal number of non-attacking rooks or
queens that can be placed on a polyomino. In one of our main results, Theorem
2 which answers Question 3 in [2], we prove that the maximal independent rook
domination problem on $d$-polycubes is NP-complete for $d\geq 3$. We also
answer the same question for queens in Theorem 1, proving that the problem is
NP-complete for $d\geq 3$.

To put our results in context, in Tables 1, 2 and 3, we collect what is known
and what is conjectured for the problems of minimal (independent and non-
independent) and maximal (independent) domination for rooks and queens. We
hope that the information in these tables will help researchers in the field
to avoid imprecision on statements about the complexity of these problems and
that it will also inspire further research. In what follows, we give precise
definitions and statements of our main results.

Max Independent Domination Problems
---
Boards |  Rooks (non-attacking) |  Queens (non-attacking)
square completion |  P (trivial) |  NP-complete [9]
all polyominoes |  P [2, Thm 12],
completion P [2, Thm 13] |  NP-complete? (Conjecture 17)
$d$–polycubes $d\geq 3$ |  NP-complete (Thm 2) |  NP-complete (Thm 1)
Table 1. Maximal independent domination problems. Min Rook Domination Problems
---
Boards |  attacking |  non-attacking (independent)
square completion |  P (trivial) |  P (trivial)
all polyominoes |  NP-hard [2, Thm 3],
NP-complete (Thm 3) |  NP-hard [2, Lem. 14],
NP-complete (Thm 3)
$d$–polycubes $d\geq 3$ |  NP-hard [2, Thm 3],
NP-complete (Thm 3) |  NP-complete (Thm 3)
Table 2. Minimal domination problems for rooks. Min Queen Domination Problem
---
Boards |  attacking |  non-attacking (independent)
square completion |  NP-complete? (Conjecture 23) |  NP-complete (Corollary 24)
all polyominoes |  NP-hard [2, Thm 4],
NP-complete (Thm 4) |  NP-complete (Thm 4)
$d$–polycubes $d\geq 3$ |  NP-hard [2, Thm 4],
NP-complete (Thm 4) |  NP-complete (Thm 4)
Table 3. Minimal domination problems for queens.

###  1.1. Main Results

We now review the main results of this paper and the relevant definitions.

######  Definition.

Let $d\geq 2$. A $\mathbf{d}$–polycube is a finite union of unit cubes of the
regular cubic tessellation of $\mathbb{R}^{d}$ with an interior that is
connected (notice that polycubes could have holes or cavities). A $2$–polycube
is also known as a polyomino.

The problems we are considering resemble art gallery problems on polyominoes
(see [4]) with guards having special guarding vision, or attacking powers,
related to chess pieces. We now give a precise definition of the attacking
powers of rooks and queens on polycubes. For this purpose, we can imagine the
$d$–cubes of the $d$–polycube centred at the points of $\mathbb{Z}^{d}$.

######  Definition (Rook attacking powers).

Suppose a rook is at $(0,\ldots,0)$ in a $d$–polycube $P$. It guards this
point. In addition, for each point that has all its coordinates $0$ except for
one coordinate $\pm 1$, we say that the $d$–dimensional rook guards or attacks
tiles which have coordinates given by all natural-number multiples of this
point such that all the smaller natural-number multiples are tiles of $P$.

######  Definition (Queen attacking powers [2]).

A $d$–dimensional queen placed in a $d$–polycube $P$ at the origin in the
cubic tessellation of $\mathbb{R}^{d}$ can attack all tiles of $P$ with
coordinates points equal to 0 or $\pm 1$ and all natural-number multiples of
such points, as long as all the smaller natural-number multiples of the points
are tiles of $P$.

Let us note that there are $d$ directions for rooks and $(3^{d}-1)/2$ for
queens, as can be seen by placing the piece at the center of a $d$-hypercube.
Figure 1 illustrates these definitions for $d=3$.

Another important remark is that the rook’s or queen’s line of attack ends
when it crosses outside the polycube.

Figure 1. Possible movement (in green) of a rook and a queen centred in a
$3\times 3\times 3$ cube. For the queen, a $5\times 5$ level is put at the top
of the cube. Find 3D models corresponding to these structures at
https://skfb.ly/oz8tJ and https://skfb.ly/oz8tn, respectively.

Our first two main results study the computational complexity of a class of
maximal independent domination problems on polycubes.

######  Theorem 1.

Solving the maximal non-attacking queen domination problem on $d$–polycubes is
NP-complete for $d\geq 3$.

######  Theorem 2.

Solving the maximal non-attacking rook domination problem on $d$–polycubes is
NP-complete for $d\geq 3$.

It was proven that, for $d=2$, there is a polynomial-time algorithm that
solves maximal domination for non-attacking rooks [2, Thm 12]. The passage to
three dimensions is the crucial point at which the complexity of this problem
changes.

We also prove NP-completeness of the minimal domination problem for non-
attacking and attacking queens and rooks, extending [2, Thms 3,4].

######  Theorem 3.

Solving the minimal domination problem for attacking and for non-attacking
rooks on $d$–polycubes for $d\geq 2$ is NP-complete.

######  Theorem 4.

Solving the minimal domination problem for attacking and for non-attacking
queens on $d$–polycubes for $d\geq 2$ is NP-complete.

We review the structure of the paper. In Section 2, we present the relevant
definitions and prove Theorems 3 and 4. In Section 3, we prove Theorems 1 and
2. In Section 4, we study domination problems polyominoes and convex
polyominoes, and state a series of open questions and conjectures. In Section
5, we present a computational model, and with it, we compute some new values
of domination numbers for classical problems. Finally, also in Section 5, we
give a brief description of the video game on chess domination on polyominoes
that we have created using the Godot game engine [20]. The game can be played
at this link: https://www.erikaroldan.net/queensrooksdomination. The software
developed and implemented in the course of this research is publicly available
on GitHub [17].

##  2\. Preliminaries

We now define the two problems we will consider.

######  Definition (Independent rook domination).

We say that an instance of the non-attacking rook set problem for
$d$–polycubes is a pair $(P,m)_{d}^{R}$, where $P$ is a $d$–polycube and $m$
is a positive integer. The problem asks whether there exists a non-attacking
configuration of $m$ rooks placed in $P$ that dominates $P$.

######  Definition (Independent queen domination).

We say that an instance of the non-attacking queen set problem for
$d$–polycubes is a pair $(P,m)_{d}^{Q}$, where $P$ is a $d$–polycube and $m$
is a positive integer. The problem asks whether there exists a non-attacking
configuration of $m$ queens placed in $P$ that dominates $P$.

We will study the complexity class of the maximal and minimal $m$ possible for
both instances $(P,m)_{d}^{Q}$ and $(P,m)_{d}^{R}$. To prove NP-complexity, we
follow the usual scheme. To help readers follow our proofs, we briefly review
the steps here. We first show the problem is in the class NP by showing that
verifying a solution is done in polynomial time. It is done in Lemma 6 below.

We then exhibit a polynomial reduction from a known NP-complete problem. This
will be the purpose of Section 3. In our case, we will begin from the
following restriction of planar sat.

######  Definition ($\mathrm{P3SAT}_{3}$ [7]).

Given a set of Boolean variables, or literals, $x_{i}$ satisfying a set of
clauses of the form $x_{i_{1}}\lor x_{i_{2}}$ or $x_{i_{1}}\lor x_{i_{2}}\lor
x_{i_{3}}$, we say it is planar if the bipartite graph constructed by two sets
of vertices given by the variables and the clause, and by edges drawn by
linking a literal $x$ and a clause $C$ if $C$ contains $x$ or $\overline{x}$
is planar. The problem of determining if all the clauses are satisfied is
called planar 3sat. If, furthermore, each $x_{i}$ appears exactly in three
clauses we call the problem planar 3sat with exactly 3 occurrences per
variable, and it is denoted $\mathrm{P3SAT}_{3}$.

Examples of such instance are given in Figures 2 and 14.

######  Remark.

It was proven to be NP-complete [7] with the further restriction that each
variable appears once positively and twice negatively. This was the problem
used for the proofs in [2]. We do not require this additional restriction.

######  Proposition 5 ([7]).

The problem $\mathrm{P3SAT}_{3}$ is NP-complete.

The reduction will proceed by introducing gadgets to encode variables,
connections and clauses as maximal independent rooks or queens set problems on
polycubes. The main difficulty will reside in showing that the size of the
polycubes created is polynomially bounded, and that the algorithm to do so
runs in polynomial time.

$c_{3}=\overline{x_{1}}\lor
x_{3}\lor\overline{x_{4}}$$x_{3}$$x_{1}$$c_{1}=x_{1}\lor
x_{2}\lor\overline{x_{3}}$$c_{2}=\overline{x_{1}}\lor\overline{x_{2}}\lor
x_{4}$$x_{2}$$x_{4}$$c_{4}=\overline{x_{2}}\lor\overline{x_{3}}\lor\overline{x_{4}}$
Figure 2. This instance of $\mathrm{P3SAT}_{3}$ has clauses $c_{1}=x_{1}\lor
x_{2}\lor\overline{x_{3}}$, $c_{2}=\overline{x_{1}}\lor\overline{x_{2}}\lor
x_{4}$, $c_{3}=\overline{x_{1}}\lor x_{3}\lor\overline{x_{4}}$, and
$c_{4}=\overline{x_{2}}\lor\overline{x_{3}}\lor\overline{x_{4}}$. It is
satisfiable; for instance, it is satisfied by
$(x_{1},x_{2},x_{3},x_{4})=(1,0,1,0)$. The example is taken from Iwamoto and
Kume’s Figure 8 [13].

Let us begin our proof of complexity by showing that the verification of a
candidate for domination and independence can be done in polynomial time.

######  Lemma 6.

Verifying that a placement of rooks or a placement of queens dominate a
$d$–polycube and verifying that a placement of queens or rooks that are not
attacking each other can be done in polynomial time.

###### Proof.

Taking each piece and checking all the cubes it guards can be done in
polynomial time. This will then tell if the pieces are independent and if all
the cubes of the polycube are guarded. ∎

We are now ready to end this section by giving the proofs of Theorem 3 and
Theorem 4.

###### Proof of Theorem 3.

By Lemma 6, we know that verifying the validity of a given set of rooks for
the minimal domination problem for attacking and non-attacking rooks can be
checked in polynomial time, and so the problems are in the class NP. The proof
of [2, Thm 3] for attacking rooks gives that it is NP-hard, so it is in fact
NP-complete.

Finally, we note that the gadgets used in the proof of [2, Thm 3] also hold
true for non-attacking rooks, thus the domination problem for non-attacking
rooks is also NP-complete. ∎

###### Proof of Theorem 4.

That a set of queens guards a polycube and that they do not attack each other
can be checked in polynomial time by Lemma 6, so the minimal domination
problem for attacking and non-attacking queens is in the class NP. The proof
of [2, Thm 4] also holds for non-attacking queens since the gadgets only use
non-attacking setups, and so it proves that both problems are NP-hard. Hence,
they are NP-complete. ∎

##  3\. Complexity of the domination problems

This section is dedicated to prove Theorems 1 and 2. We will begin with the
rooks and proceed to the queens.

###  3.1. NP-Completion of maximal independent domination of rooks on
polycubes

We first introduce the variable gadget for non-attacking rooks in Figure 3.
There are two ways to place the maximal number of rooks on the gadget.

TTFF Figure 3. Variable gadget with rooks; when true, 2 additional rooks go on
the dark cubes (T), and when false, 2 go on the light red cubes (F). Find a 3D
model corresponding to this structure at https://skfb.ly/oz8tZ.

######  Lemma 7.

The maximal number of dominating non-attacking rooks for the polycube of
Figure 3 is 6, and there are only two ways to do so.

###### Proof.

The middle holed section can be guarded either by four non-attacking rooks in
a cross pattern, which would dominate the full polyomino, or by two rooks in
opposite corners, which then require four rooks on the remaining unguarded
cubes. Thus at most 6 non-attacking rooks can guard the polyomino.

Four rooks are placed on the protruding corners, and two additional ones can
be placed either both on the dark T cubes, and then we say the variable has
truth value true; or both on the red F cubes, and then we say the variable has
truth value false. ∎

FTFT (a) The connection gadget.
https://skfb.ly/ozJZn

TTFFFTFT (b) Connection gadget on a variable gadget.
https://skfb.ly/ozV7S

Figure 4. The connection gadget for rooks, values are transmitted along T and
F cubes for true and false respectively.

To transmit the signal, we will need to use the third dimension. Note,
however, that the gadgets will never need to exceed 9 cubes in height. Let us
introduce the connection gadgets in Figure 4. It has a maximal covering number
by rooks of 6. It is placed on the variable gadget on one of the NW or SW
corners of the variable gadget, with the protruding T and F tiles on F and T
tiles respectively. According to the variable gadget truth value, the maximal
independent covering of rooks on the connection gadget will add rook on the T
or F tiles.

######  Lemma 8.

The signal of the variable gadget is propagated by the connection gadget of
Figure 4. Furthermore, the connection gadget can be used to turn corners.

###### Proof.

As we connect to the variable gadget placing a red F tile on a dark T tile,
and a dark T tile to a red F tile, the truth value of the variable gadget will
influence where we can place the maximal 6 rooks on the connection gadget.
First, three rooks are placed on the corners. Then if the variable has value
true, the remaining rooks are placed on T tiles. Precisely, one on the T tile
above the variable gadget, one on the cube just behind the F cube placed above
the variable gadget and one on the top T cube of the connection gadget. This
process can be repeated on the connection gadget to place a variable tile at
the top. One can also keep on the same level by doing the process another
time. What we described is viewed on the YZ-axis in Figure 5.

TTTTTTTFFFFFFF Figure 5. 2D front view of the connection, 11 rooks are not
shown.

Finally, we can go around corners by rotating in space the connection gadget,
always propagating the signal, and duplicating it if needed, as shown in
Figure 6. The height we need for this is 9 cubes. ∎

(a) Turning the corner with connection gadgets.
https://skfb.ly/ozUTt

(b) Duplicating the signal with connection gadgets.
https://skfb.ly/ozUSO

Figure 6. Connection gadgets propagating and duplicating the signal.

We introduce the clause gadgets in Figures 7. The clause gadget is created as
follows. Let $c$ be a clause and suppose the literal gadgets are aligned on
the NE axis. We place a connector  on top of the NW red F tile in the variable
$x$ if $x$ appears as $x\in c$ and on the NW gray T tile if it is negated. The
connectors are joined together by alternating top  and left  nodes in a line.
We denote $\ell_{R}(c)$ the length of the clause gadget associated with $c$

(1) $\ell_{R}(c):=|\\{\text{nodes and connectors in $c$}\\}|.$

Figure 8 presents one example.

######  Lemma 9.

Let $c$ be a clause and $K(c)$ be the clause gadget associated with it. A
maximal rook placement on $K(c)$ contains $\ell_{R}(c)$ rooks if $c$ evaluates
to false and $\ell_{R}(c)+1$ if it evaluates to true.

TT (a) Clause gadget linked to the clauses with two positive or two negated
literals.
https://skfb.ly/ozVqt

TT (b) Clause gadget linked to the clauses with one positive and one negated
literals.
https://skfb.ly/ozVqH

Figure 7. The clause gadgets with two literals, all light green cubes are
guarded, and one additional rook might be placed on one of the two dark T
cubes.

###### Proof.

Construct the clause gadget $K(c)$ from the clause $c$. On each top and left
nodes there can be one rook, and there can be at most two rooks on the
connectors. However, the connectors are all placed on a line and so only one
can have two rooks. There are one node or connector for each tile, thus
$\ell_{R}(c)$. By construction as $K$ is placed atop the variable, there can
only be an extra rook if the clause evaluates to true. For example, Figure 8
presents a clause gadget $K(c)$ of length 13 for the clause $c=x_{1}\lor
x_{2}\lor x_{3}$. No extra rook can be placed if all literals are false. ∎

TTFFTTFFTTFFTTT$\mathbf{x_{1}}$$\mathbf{x_{2}}$$\mathbf{x_{3}}$ Figure 8. The
clause gadget of the clause
$\mathbf{x_{1}}\lor\mathbf{x_{2}}\lor\mathbf{x_{3}}$.
https://skfb.ly/ozV8L

######  Proposition 10.

From an instance $C$ of $\mathrm{P3SAT}_{3}$, it is possible to construct a
polycube $P_{3}^{R}(C)$ of polynomial size in polynomial time.

###### Proof.

The polycube $P_{3}^{R}(C)$ is constructed from $C$ by replacing each variable
of $C$ by the variable gadget of Figure 3, and by using an amount of
connection gadgets of Figure 4 to transmit their signal or its inverse to
clause gadgets that will combine their outputs.

The construction of connection gadget has a height of at most 9 for a corner
or a split, and the clause gadget has a height of 3, so the size of the
constructed polycube is inside a rectangular prism of height 12 at most, hence
still of order $O(n^{2})$. We can thus obtain a polynomial-time algorithm to
convert an instance $C$ to a polycube $P_{3}^{R}(C)$ whose size is bounded
polynomially again by following the idea of the proof of [2, Lem. 8]. ∎

######  Lemma 11.

Let $C$ be an instance of $\mathrm{P3SAT}_{3}$, and let $P_{3}^{R}(C)$ be its
associated polycube. Let

(2) $m_{R}:=6n_{var}+6n_{connect}+\sum_{c\in\\{\text{clauses of
}C\\}}\ell_{R}(c),$

where $n_{var}$ is the number of variables, $n_{connect}$ is the total number
of connection gadgets, $\ell_{R}(c)$ is the length of the clause $c$, and
$n_{clause}$ is the total number of clauses. There exists a non-attacking rook
set of size $m_{R}+n_{clause}$ for the polycube $P_{3}^{R}(C)$ if and only if
$C$ is satisfiable.

###### Proof.

Consider a truth assignment that satisfies all the clauses in $C$. We transfer
the assignment on the polycube $P_{3}^{R}(C)$ of Proposition 10. Each variable
can have at most 6 rooks placed on them regardless of their value, so there
are 6 rooks per variable gadget.

We connect the variable gadgets to the clause gadgets and each of them have a
maximal number of 6 rooks regardless of the signal they transfer, so there can
be 6 rooks per connection gadget.

Since it is a truth assignment, all clauses are satisfied. Lemma 9 implies
that one additional non-attacking rook per clause can be placed.

The total number of non-attacking rooks that can be placed on $P_{3}^{R}(C)$
is then $m_{R}+n_{clause}$, proving that an instance $C$ is transformed into
an independent rook domination problem
$(P_{3}^{R}(C),m_{R}+n_{clause})_{3}^{R}$.

If a set of $m_{R}+n_{clause}$ non-attacking rooks dominating $P_{3}^{R}(C)$
exists and since it is not possible to increase the number of rooks in the
variable nor in the connection gadgets, there must be $\ell_{R}(c)+1$ rooks
placed on each clause $c$ gadget. From Lemma 9, it follows that all the
corresponding clauses evaluate to true, meaning $C$ is satisfied. ∎

We can now prove Theorem 2.

###### Proof of Theorem 2.

Lemma 6 implies that the problem is in the class NP since verifying a solution
can be done in polynomial time.

Let $C$ be an instance of $\mathrm{P3SAT}_{3}$. We obtain a polynomially-sized
polycube $P_{3}^{R}(C)$ in polynomial time by Proposition 10. Lemma 11 implies
that the non-attacking rook set problem for
$(P_{3}^{R}(C),m_{R}+n_{clause})_{3}^{R}$ is equivalent to the
$\mathrm{P3SAT}_{3}$ problem for $C$, which is NP-complete by Proposition 5,
and so the non-attacking rook set problem for $3$–polycube is NP-complete. As
$3$–polycubes are contained in $d$–polycubes for $d\geq 3$, the non-attacking
rook set problem for $d$–polycubes is also NP-complete. ∎

###  3.2. NP-completion of maximal independent domination of queens on
polycubes

We now turn to the proof of Theorem 1. It will proceed in a similar fashion as
the previous section, with different gadgets. The first component of the proof
is the literal gadget obtained as a concatenation of four smaller elements
identical to the rook variable gadgets; see Figure 9.

TTTTFFFF (a) One element.

TTTTTTTTTTTTFFFFFFFFFFFF (b) Literal gadget for 3D queens from four elements.

Figure 9. Construction of the literal gadget for 3D queens. Maximally 12
queens can be placed in two different ways.

######  Lemma 12.

The gadget of Figure 9(b) has two states of maximal domination by 12 queens.

###### Proof.

The proof proceeds in a similar fashion as for the rook case. It can be
checked by the solver [18] introduced in Tool 25 since the gadget is two-
dimensional. ∎

Contrary to the rook gadgets, the number of queens will not remain constant on
the connected literal gadget. This means that we must analyse carefully the
propagation of the signal via the connection gadget presented in Figure 10.

TTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTTFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF$\uparrow$$\rightarrow$Lit.
Figure 10. Top view of the connection gadget (in the dotted teal box)
propagating the signal of a variable gadget (in the full orange box). There is
one pair of elements with each 4 neighbours in the middle (in a dashed black
box)

######  Lemma 13.

The signal of the variable gadget is propagated by the connection gadget of
Figure 10. The number of queens needed to guard the gadget is given by
$4n_{4neigh}+5n_{3neigh}+6n_{2neigh}$, where $n_{ineigh}$ is the number of
pairs of tiles with $i$ neighbours. Furthermore, the connection gadget can be
used to turn corners.

###### Proof.

We begin by counting the number of queens on the gadgets. We notice first that
the basic elements come in pairs by construction. The members of the pair can
either both have 2, 3 or 4 neighbours. Then a local analysis shows that the
pair will have maximally 6, 5 or 4 queens respectively. The three situations
can be seen in Figure 10, where the one pair with 4 neighbours has been
highlighted. ∎

Finally, to evaluate clauses, gadgets similar to those of Figures 7 and 8 are
used. Their construction proceeds as follows. Let $c$ be a clause with three
literals (respectively two). Suppose there are three literal gadgets
$x_{1},x_{2},x_{3}$ (respectively two $x_{1},x_{2}$) on a line. Then the
extremal nodes form a sequence of red (false) and gray (true) tiles. If the
literal $x_{i}$ appears as $x_{i}$ in $C$, we place a connector  on the red
(F) tile, and if it appears as $\overline{x}_{i}$ in $C$, we place the
connector on the gray (T) tile. We then join the connector by alternating top
and left  nodes. They form a clause gadget—see Figure 12. The length of the
clause gadget is denoted $\ell_{Q}(c)$ and is defined similarly as the length
$\ell_{R}$ of the rook gadget in (1). The process is illustrated in Figure 11.

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