Asymptotic growth rate of square grids dominating sets: a symbolic dynamics approach

TL;DR

This paper establishes the existence of an asymptotic growth rate for dominating sets on large rectangular grids and provides algorithms and bounds for computing these rates.

Contribution

It introduces a symbolic dynamics approach to prove growth rate existence and offers algorithms to compute and bound these rates for various dominating set variants.

Findings

Existence of asymptotic growth rates for dominating sets on large grids

Algorithms for computing growth rates of dominating sets

Bounds on growth rates obtained via computer programs

Abstract

In this text, we prove the existence of an asymptotic growth rate of the number of dominating sets (and variants) on finite rectangular grids, when the dimensions of the grid grow to infinity. Moreover, we provide, for each of the variants, an algorithm which computes the growth rate. We also give bounds on these rates provided by a computer program.

Equations

$$\gamma(G_{n,m})=\Bigl{\lfloor}\frac{(n+2)(m+2)}{5}\Bigr{\rfloor}-4.$$

$$X=\left\{x\in {\mathcal{A}}^{{\mathbb{Z}}^d}:\forall \text{u}\in {\mathbb{Z}}^d,x_{|\text{u}+\mathbb{U}}\notin \mathcal{F}\right\}.$$

$$({\sigma }^{\text{u}}x{)}_{\text{v}}=x_{\text{v}+\text{u}}\text{.}$$

$$h(X)=\underset{n}{\inf }\frac{{\log }_2(N_n(X))}{n^d}.$$

$$h(X)=\underset{n}{\lim }\frac{{\log }_2(N_n(X))}{n^d}.$$

$$h(X)=\underset{n,m}{\lim }\frac{{\log }_2(N_{n,m}(X))}{nm}.$$

$$\mathcal{N}(\mathbb{U})=\bigcup _{\text{u}\in \mathbb{U}}\left(\text{u}+[[-1,1{]]}^2\right).$$

$${\mathbb{B}}_{n,m,k}={\mathcal{N}}^k([[1,n]]\times [[1,m]])\setminus {\mathcal{N}}^{k-1}([[1,n]]\times [[1,m]]).$$

$${\mathbb{B}}_{m,n,k+1}$$

$$\frac{1}{2^{6(n+m)}}{N}_{n,m}(X^M)\le M_{n,m}\le N_{n,m}(X^M).$$

$$\frac{1}{2^{8(m+n)}}{N}_{n,m}(X^{MT})\le M{T}_{n,m}\le N_{n,m}(X^{MT}).$$

$$\boxed{D_{n,m}={\nu }_D^{nm+o(nm)}}$$

$$-\frac{12(m+n)}{nm}+\frac{{\log }_2(N_{n,m}(X^M))}{nm}\le \frac{{\log }_2(M_{n,m})}{nm}\le \frac{{\log }_2(N_{n,m}(X^M))}{nm}.$$

$$\frac{{\log }_2(M_{n,m})}{nm}\to h(X^M).$$

$$\delta (\mathbb{U},\mathbb{V})=\underset{\text{u}\in \mathbb{U}}{\min }\underset{\text{v}\in \mathbb{V}}{\min }||\text{v}-\text{u}|{|}_{\infty }.$$

$$⋮$$

$$p$$

$$⋮$$

$$h(X)={\log }_2\underset{n}{\lim }||M^n|{|}^{1/n}.$$

$$D_{n,m}=2^{h(X^{D,m})\cdot n+o(n)}$$

$$\phi$$

$$h(X^{D,m})=h(\phi (X^{D,m})).$$

$$D_{n,18k}\ge D_{n,18}^k=2^{h(X^{D,18})\cdot kn+k\cdot o(n)},$$

$$\underset{n,m}{\lim }\frac{{\log }_2(D_{n,m})}{nm}=\underset{n,k}{\lim }\frac{{\log }_2(D_{n,18k})}{18nk}\ge h(X^{D,18}).$$

$$D_{n,m}\le D_{n,m}^{\ast }.$$

$$D_{n,18k}^{\ast }\le (D_{n,18}^{\ast }{)}^k=2^{h(X_{\ast }^{D,18})\cdot km+k\cdot o(n)}.$$

$$h(X)\le h(X_{\ast }^{D,18}).$$

Full text

ENSL, Univ Lyon, UCBL, LIP, MC2, F-69342, LYON Cedex 07, France

silvere.gangloff@ens-lyon.fr, alexandre.talon@ens-lyon.fr

1 Introduction

A dominating set $S$ of a graph $G$ is a subset of its vertices such that any vertex not in $S$ is connected to a vertex in $S$. The dominating number $\gamma(G)$ of the graph is the minimum cardinality of a dominating set. These notions appear in practical problems, related to robotics and networks constructions. Some decidability results are known: for instance, the problem $\gamma(G)\leq k$ given the integer $k$ and the finite graph $G$ is NP-complete. An important problem has been to compute exactly the dominating number for finite rectangular grids, and it was solved by D. Gonçalves, A. Pinlou, M. Rao, S. Thomassé [Gonçalves et al.], proving Chang’s conjecture, which tells that denoting $G_{n,m}$ the finite $n\times m$ rectangular grid,

[TABLE]

Another problem, which is still open, is to compute the number of dominating set of graphs. Some formulas are known, such as a relation between this number and the number of complete bipartite subgraphs of the complement of $G$ [Heinrich Tittmann].

In the present text, we are interested in the asymptotic growth rate of the number of dominating sets, on the finite rectangular grids $G_{n,m}$, when $n$ and $m$ grow to infinity. We also study this problem for the total domination, the minimal domination, and the minimal total domination. The text is organised as follows:

In Section 2, we define the various notions of dominating sets on (finite or infinite) graphs we have just mentioned, and prove local characterisations of these sets. 2. 2.

In Section 3 we associate, to each of these notions of dominating sets, a symbolic dynamical system called subshift of finite type, which consists in a set of colourings of the infinite grid $\mathbb{Z}^{2}$. Comparing the number of dominating sets on finite grids and the number of patterns which appear in configurations of the corresponding subshift, we prove the existence of a growth rate and show that it is equal to the entropy of the dynamical system. 3. 3.

In Section 4, we define the block-gluing property; any subshift of finite type that is block-gluing is guaranteed to have an entropy which is computable in an algorithmical sense. We then prove that the various domination subshifts defined in Section 3 are block gluing. This fact provides an algorithm which computes approximations of the growth rate given the desired precision in the input. 4. 4.

In Section 5, we provide some bounds for the growth rates obtained by a computer program.

2 Notions of dominating sets of square grids

2.1 Definitions

In the following, for a graph $G=(V,E)$, we will say that two vertices $u,v$ in $V$ are neighbours or connected when the edge $\{u,v\}$ is in $E$. For all $n\geq 1$, we will denote by $G_{n,m}$ the finite square grid graph of size $n\times m$.

Definition 1**.**

Let $G=(V,E)$ be a graph, $S$ a subset of $V$ and $v\in V$. We say that $v$ is dominated by $S$ if $v$ has a neighbour in $S$.

Definition 2**.**

Let $G=(V,E)$ be a graph. A subset $S\subseteq V$ is said to be a dominating set of the graph $G$ when every $v\in V\setminus S$ is dominated by $S$. It is said to be a minimal dominating set of $G$ when it is a dominating set of $G$ and for all $v\in S$, $S\setminus\{v\}$ is not dominating. It is said total dominating when for all $v$ in $V$, $v$ has a neighbour in $S$.

Definition 3**.**

A subset $S\subset V$ is said to be minimal total dominating when it is total dominating and for all $v\in S$, $S\setminus\{v\}$ is not total dominating.

Notice that a minimal dominating set (resp. minimal total dominating set) is a dominating set (resp. total dominating set) which is inclusion-wise minimal. These notions are illustrated in Figure 1.

Definition 4**.**

When a dominating set $S$ of a graph $G$ is fixed, a vertex is called a dominant element of $G$ when it is in $S$, and a dominated element when it has a neighbour in $S$. A neighbour $w$ of a dominant element $v$ is said to be a private neighbour of $v$ when $v$ is the only neighbour of $w$ in the set $S$.

2.2 Local characterisations

In this section, we recall, and for completeness prove, local characterisations of the notions of dominating sets. This means that one can check if a set $S$ is dominating (or minimal dominating, etc) by checking, for each vertex, whether or not this vertex and its neighbours are in $S$

Fact 1**.**

Let $S$ be a set of vertices of a graph $G=(V,E)$. Then for all $v$ and $w$ such that $w$ is not a neighbour of $v$, $w$ is dominated by $S$ if and only if it is dominated by $S\setminus\{v\}$.

Definition 5**.**

Let $S$ be a set of vertices of a graph $G=(V,E)$. We say that a dominant element is isolated in $S$ when it has no neighbours in $S$.

Proposition 1**.**

Let $S$ be a dominating set of a graph $G=(V,E)$. $S$ is minimal dominating if and only if any of its elements is isolated in $S$ or has a private neighbour not in $S$.

Proof.

• •

$(\Rightarrow)$: Let us assume that $S$ is minimal dominating, and fix $v\in S$. From Fact 1 and by definition of a minimal dominating set, any $w$ which is not in the neighbourhood of $v$ is dominated by $S\backslash\{v\}$. Since $S\setminus\{v\}$ is not dominating, it means that:

$v$ is not dominated by $S\setminus\{v\}$, which means that $v$ is isolated in $S$, 2. 2.

or there exists some $u\notin S$ neighbour of $v$ which is not dominated by $S\setminus\{v\}$, hence $u$ is a private neighbour of $v$ which is not in $S$.

• •

$(\Leftarrow)$: Conversely, let us fix some dominating set $S$ such that every $v\in S$ has a private neighbour not in $S$ or is isolated. Fix some $v\in S$. If it has a private neighbour $u$, then $u$ is not dominated by $S\setminus\{v\}$, and thus $S\setminus\{v\}$ is not dominating. If it has no private neighbours, then it is isolated. This means that $v$ is not dominated by $S\setminus\{v\}$, therefore the set is not dominating. In both cases, we conclude that $S$ is minimal dominating.

∎

With a similar proof, we obtain the following:

Proposition 2**.**

A total dominating set $S$ of a graph $G$ is minimal total dominating if and only if any $v\in S$ has a private neighbour.

In the following, we will use the following notations:

Notation 1**.**

In the following, for all integers $n,m$, we denote by $D_{n,m}$, $M_{n,m}$ and $T_{n,m}$ respectively the number of dominating sets of the grid $G_{n,m}$, the number of its minimal dominating sets and the number of its minimal total dominating sets.

3 From dominating sets to subshifts of finite type

In this section, we introduce the notion of subshift of finite type on a regular grid (see Section 3.1), which consists in sets of possible colourings of the grid avoiding some forbidden patterns. After presenting some examples which are the subshifts counterparts of various notions of domination in Section 3.2, we use the well-known fact that the entropy of a subshift can be expressed as a limit to prove the existence of an asymptotic growth rate of the number of dominating sets in Section 3.3.

3.1 Subshifts of finite type

Definition 6**.**

Let $\mathcal{A}$ be a finite set, and $d\geq 1$ integer. A pattern $p$ on alphabet $\mathcal{A}$ is an element of $\mathcal{A}^{\mathbb{U}}$ for some finite $\mathbb{U}\subset\mathbb{Z}^{d}$. The set $\mathbb{U}$ is called the support of $p$, and is denoted $\text{supp}(p)$. Informally, it is the location of $p$ in the grid $\mathbb{Z}^{d}$.

Notation 2**.**

For a configuration $x=(x_{\textbf{u}})_{\textbf{u}\in\mathbb{Z}^{d}}$ of $\mathcal{A}^{\mathbb{Z}^{d}}$ (resp. a pattern $p\in\mathcal{A}^{\mathbb{U}}$ for some $\mathbb{U}\subset\mathbb{Z}^{d}$), we denote by $x_{|\mathbb{V}}$ the restriction of $x$ to some subset $\mathbb{V}\subset\mathbb{Z}^{d}$ (resp. the restriction of $p$ to $\mathbb{V}\subset\mathbb{U}$).

Definition 7**.**

Let $\mathcal{A}$ be a finite set, and $d\geq 1$ integer. A $d$-dimensional subshift of finite type (SFT) on alphabet $\mathcal{A}$ is a subset of $\mathcal{A}^{\mathbb{Z}^{d}}$ defined by a finite set of forbidden patterns. Formally, a subset $X$ of $\mathcal{A}^{\mathbb{Z}^{d}}$ is a subshift of finite type when there exist some finite sets $\mathbb{U}\subset\mathbb{Z}^{d}$ and $\mathcal{F}\subset\mathcal{A}^{\mathbb{U}}$ such that:

[TABLE]

The elements of $\mathcal{F}$ are called the forbidden patterns. When the set of forbidden patterns is fixed, a pattern which does not contain any forbidden pattern is called locally admissible.

Notation 3**.**

Let us denote by $\sigma$ the $\mathbb{Z}^{d}$-shift action on $\mathcal{A}^{\mathbb{Z}^{d}}$ defined such that for all $\textbf{u},\textbf{v}\in\mathbb{Z}^{d}$,

[TABLE]

Informally, $\sigma$ acts on a configuration by translating it by the vector u.

Definition 8**.**

For a SFT $X$ a globally admissible pattern of size $\llbracket 1,n\rrbracket^{d}$ is some $p\in\mathcal{A}^{\llbracket 1,n\rrbracket^{d}}$ which appears in a configuration of $X$, that is when $x_{|\llbracket 1,n\rrbracket^{d}}=p$. When $d=2$, we extend the definition to patterns $p\in\mathcal{A}^{\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket}$ when there exists a configuration $x$ of $X$ such that $x_{|\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket}=p$.

Notation 4**.**

For a subshift of finite type $X$, we denote by $N_{n}(X)$ the number of globally admissible patterns of size $\llbracket 1,n\rrbracket^{d}$. When $d=2$, we extend the notation and denote by resp. $N_{n,m}(X)$ the number of globally admissible patterns of size $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$.

Definition 9**.**

The topological entropy of a subshift of finite type is the number

[TABLE]

The following three lemmas are well known (see for instance [Lind Marcus]).

Lemma 1**.**

The infimum in the definition of $h(X)$ is in fact a limit:

[TABLE]

Definition 10**.**

A conjugation between two $d$-dimensional subshifts of finite type $X$ and $Z$ is an invertible map $\varphi:X\rightarrow Z$ such that for all $\textbf{u}\in\mathbb{Z}^{d}$ and $x\in X$, $\varphi(\sigma^{\textbf{u}}.x)=\sigma^{\textbf{u}}.\varphi(x)$. In this case, $X$ and $Z$ are said to be conjugated.

Lemma 2**.**

If two subshifts of finite type $X$ and $Z$ are conjugated, then $h(X)=h(Z)$.

Lemma 3**.**

Let $X$ be a bidimensional subshift of finite type. Then:

[TABLE]

3.2 Domination subshifts

In this section, the alphabet is \mathcal{A}_{0}=\left\{\leavevmode\hbox to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.575,0.575,0.575}\definecolor[named]{pgfstrokecolor}{rgb}{0.575,0.575,0.575}\pgfsys@color@gray@stroke{0.575}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0.575}\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{0.575,0.575,0.575}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right\}, and $d=2$.

Definition 11**.**

The domination (resp. minimal domination, total domination and minimal total-domination) denoted by $X^{D}$ (resp. $X^{M}$, $X^{T}$ and $X^{MT}$), is the set of elements $x$ of $\mathcal{A}_{0}^{\mathbb{Z}^{2}}$ such that \{\textbf{u}\in\mathbb{Z}^{2}:x_{\textbf{u}}=\leavevmode\hbox to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\color[rgb]{0.575,0.575,0.575}\definecolor[named]{pgfstrokecolor}{rgb}{0.575,0.575,0.575}\pgfsys@color@gray@stroke{0.575}\pgfsys@invoke{ }\pgfsys@color@gray@fill{0.575}\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{0.575,0.575,0.575}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@fill\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{{}}{} {}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\} is a dominating (resp. minimal dominating, total dominating and minimal total dominating) set of the infinite square grid $\mathbb{Z}^{2}$. In all these cases, a configuration $x$ of the subshift is called a dominated configuration. We also say that u is a dominant position of the configuration $x$ when $x_{\textbf{u}}$ is grey. Likewise, a private neighbour is still a position which is dominated by exactly one dominant position.

Property 1**.**

The sets $X^{D}$, $X^{M}$, $X^{T}$ and $X^{MT}$ are subshifts of finite type.

Proof.

This comes from the local characterisations of each type of dominating sets [Section 2.2], which can straightforwardly be translated into forbidden patterns. ∎

3.3

Existence of an asymptotic growth rate

The dominating sets of a finite grid $G_{n,m}$ do not correspond exactly to the globally admissible patterns on the same grid of the corresponding subshifts of finite type presented in Section 3.2. Indeed, in such a pattern, the positions of the border may for instance be dominated by a position outside the pattern in a configuration in which it appears. However we will see that we can compare the number of globally admissible patterns of size $n\times m$ for $X^{D}$ (resp. $X^{M}$, $X^{T}$ and $X^{MT}$) with the number of dominating sets (resp. minimal dominating sets, total dominating set and minimal total dominating sets) of $G_{n,m}$. We use this to prove the existence of an asymptotic growth rate for the grid, equal to the entropy of the SFT.

In this section, we assimilate the set of vertices of $G_{n,m}$ to any translate of $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ and assimilate any set $S$ of vertices of a finite grid $G_{n,m}$ with the pattern $p$ of $\mathcal{A}^{\mathbb{Z}^{2}}$ on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ defined by $p_{\textbf{u}}$ being grey if and only if $\textbf{u}\in S$.

Notation 5**.**

If $\mathbb{U}$ is a subset of $\mathbb{Z}^{2}$, we define the (extended) neighbourhood of $\mathbb{U}$ as

[TABLE]

We also define, for all $n,m\geq 1$ and $k\geq 1$ the border

[TABLE]

[TABLE]

For convenience, we extend the notation to $\mathbb{B}_{n,m,0}=\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket\setminus\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$.

Lemma 4**.**

For all $n,m\geq 2$, the following inequalities hold: $N_{n-1,m-1}(X^{D})\leq D_{n,m}\leq N_{n,m}(X^{D})$.

Proof.

For all $n,m\geq 1$, any dominating set of $G_{n,m}$ can be extended into a configuration of $X^{D}$ by defining the symbol of any position outside $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ to be grey. As a consequence, any dominating set of $G_{n,m}$ is globally admissible in $X^{D}$ and thus $D_{n,m}\leq N_{n,m}(X^{D})$. 2. 2.

Any pattern of $X^{D}$ on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ can be turned into a dominating set of $\llbracket 0,n+1\rrbracket\times\llbracket 0,m+1\rrbracket$ by extending it with grey symbols. Hence we obtain the inequality $N_{n,m}(X^{D})\leq D_{n+1,m+1}$ for all $n,m\geq 1$.

∎

Using the very same arguments, we obtain the same inequality for the total domination.

Lemma 5**.**

For all $n,m\geq 2$, the following inequalities hold: $N_{n-1,m-1}(X^{T})\leq T_{n,m}\leq N_{n,m}(X^{T})$.

We then address the minimal and minimal total domination. The proofs of the following inequalities are more complex as we will see.

Lemma 6**.**

For all $n,m\geq 1$, the following inequalities hold:

[TABLE]

Proof.

Second inequality.

1. (a)

A completion algorithm of a dominating set into a configuration of $X^{M}$. Let $S$ be a minimal dominating set of $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$. Let us extend it into a configuration $x$ of $X^{M}$ using the following algorithm: successively for every $k\geq 0$, we extend the current pattern into a pattern on $\mathcal{N}^{k+1}(\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket)$ using the following operations, for all $\textbf{u}\in\mathbb{B}_{n,m,k+1}$:

1. i.

if u is a corner then $x_{\textbf{u}}$ is white; 2. ii.

if u is a neighbour of a corner in one of the vertical sides of $\mathbb{B}_{n,m,k+1}$ then $x_{\textbf{u}}$ is white; 3. iii.

for all other u, $x_{\textbf{u}}$ is grey if and only if its neighbour in $\mathcal{N}^{k}(\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket)$ is neither dominated by an element in this set, nor a dominant element.

This algorithm is illustrated in Figure 2.

2. (b)

The output obtained by repeating the algorithm is a configuration of $X^{M}$.

• •

Every position is dominated. This is verified for the positions in $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$. Outside this set, if a position in some $\mathcal{N}^{k}(\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket)$ (for $k\geq 0$) is not dominated before extending the configuration on $\mathcal{N}^{k+1}(\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket)$, then it gets dominated at this step by Rule iii and stays that way afterwards.

• •

Every dominant position is isolated or has a private neighbour.

Let us consider a dominant position u which is not isolated. If it lies in $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$, then it has a private neighbour since the pattern on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ is a minimal dominating set of $G_{n,m}$. Else, it lies in some $\mathbb{B}_{n,m,k}$ for some $k\geq 0$ and there are two cases:

• –

u** is not a corner.** Its neighbour $\textbf{v}\in\mathbb{B}_{n,m,k+1}$ is white by the application of the algorithm. Also, since its neighbours in $\mathbb{B}_{n,m,k}$ are thus dominant or dominated, their neighbours in $\mathbb{B}_{n,m,k+1}$ are white. Moreover, the neighbour of v in $\mathbb{B}_{n,m,k+2}$ is thus white. This is illustrated in Figure 3. As a consequence v is a private neighbour for u.

• –

u** is a corner.** We apply a similar reasoning.

First inequality.

1. (a)

Transformating patterns of $X^{M}$ into minimal dominating sets. Let us define an application $\phi_{n,m}$ which, to each pattern of $X^{M}$ on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$, associates a minimal dominating set of $G_{n,m}$ defined by:

1. i.

suppressing any dominant position in $\mathbb{B}_{n,m,0}$ which has no private neighbours in $G_{n,m}$ and which is dominated by an element of $G_{n,m}$; 2. ii.

changing successively any non-dominant position of $\mathbb{B}_{n,m,0}$ which is still not dominated into a dominant one; 3. iii.

successively, for every dominant position $\textbf{u}\in\mathbb{B}_{n,m,0}$ : if one of u’s neighbours v is the only private neighbour of a position w which is not isolated in $G_{n,m}$ then change w into a non-dominant position.

This Step is illustrated on Figure 4.

2. (b)

Verifying that images of $\phi_{n,m}$ are minimal dominating sets.

Let us consider a globally admissible pattern $p$ of $X^{M}$ on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$. The set $\phi_{n,m}(p)$ is a minimal dominating set of $G_{n,m}$:

• •

Any vertex of $G_{n,m}$ is dominated or dominant in $\phi_{n,m}(p)$.

Before Step ii, if a position is not dominant and not dominated, it becomes dominant during this step. Moreover, during Step iii, any position which is modified is transformed into a dominated position.

• •

Any non-isolated dominant position has a private neighbour. After applying $\phi_{n,m}$, only the positions on the border $\mathbb{B}_{n,m,0}$ may not have any private neighbour. After Step i, every

dominant position on $\mathbb{B}_{n,m,0}$ is isolated, or has a private neighbour. After Step ii, some positions may be dominant, non-isolated, and have no private neighbours. Such positions become non-dominant in Step iii. 3. (c)

For all $n,m$, the number of preimages of $\phi_{n,m}$ for any minimal dominating set of $G_{n,m}$ is bounded (roughly) by $2^{3(2n+2m)}$, since any symbol modified by the application is at distance at most 2 from $\mathbb{B}_{n,m,0}$. As a consequence, $N_{n,m}(X^{M})\leq 2^{6(n+m)}M_{n,m}$.

∎

Lemma 7**.**

For all $n$, the following bounds hold:

[TABLE]

For readability, we reproduce the structure of the proof of Lemma 6, but simplify the arguments and refer this proof.

Proof.

Second inequality.

1. (a)

A completion algorithm of dominating set into a configuration of $X^{MT}$.

Consider a minimal total dominating set of $G_{n,m}$. Any element in $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$ is dominated by an element of $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$, and any dominant element in $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$ is not isolated and has a private neighbour in $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ (which may or may not be a dominant position). Let us extend it into a configuration $x$ of $X^{MT}$ using an algorithm very similar to the one in the corresponding point in the proof of Lemma 6, but the condition in the third point is different:

1. i.

if u is a corner then $x_{\textbf{u}}$ is white; 2. ii.

if u is a neighbour of a corner in one of the vertical sides of $\mathbb{B}_{n,m,k+1}$ then $x_{\textbf{u}}$ is white; 3. iii.

for all other u, $x_{\textbf{u}}$ is grey if and only if its neighbour in $\mathcal{N}^{k}(\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket)$ is not dominated by an element in this set. 2. (b)

The result of the algorithm is a configuration of $X^{MT}$.

• •

Every position is dominated. Similar to the corresponding point in the proof of Lemma 6. This implies that no dominant positions are isolated.

• •

Every dominant position has a private neighbour.

Let us consider a dominant position u. If it is in $\llbracket 3,n-2\rrbracket\times\llbracket 3,m-2\rrbracket$, since the pattern on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$ is a minimal total dominating set of $G_{n,m}$, we know that it has a private neighbour. Else, it lies in some $\mathbb{B}_{n,m,k}$ for $k\geq 0$, or in $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$. Then there are two cases:

• –

u** is not a corner.** If it has no dominant neighbours in Let us call v its neighbour in $\mathbb{B}_{n,m,k+1}$. Note that, depending on whether or not u is dominated inside $\mathcal{N}^{k}(\mathbb{B}_{n,m,0})$, v may be white or grey. Since the neighbours of u in $\mathbb{B}_{n,m,k}$ are dominated, v’s neighbours in $\mathbb{B}_{n,m,k+1}$ are white. Finally, since v is dominated by u, its neighbour in $\mathbb{B}_{n,m,k+2}$ is white, hence v is a private neighbour for u.

• –

u** is a corner.** We apply a similar reasoning. 2. 2.

First inequality:

1. (a)

A transformation of patterns of $X^{MT}$ into dominating sets.

Let us define once again an application $\phi_{n,m}$ which, to each pattern of $X^{MT}$ on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$, associates a minimal total dominating set of $G_{n,m}$, defined in a similar way as in the corresponding point in the proof of Lemma 6, but the proof is more complex.

1. i.

suppress any dominant position on the border $\mathbb{B}_{n,m,0}$ which has no private neighbours in $G_{n,m}$. 2. ii.

Successively, for every non-corner undominated position u on the border $\mathbb{B}_{n,m,0}$, do the following:

• •

Consider the position v, neighbour of u in $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$. For each dominant position w in the neighbourhood of v, and for each dominant position w’ in the neighbourhood of w, if w is the only private neighbour of w’, then change w’ into a non-dominant position.

• •

Change v into a dominant position.

Then do the same operations for the corners of $\mathbb{B}_{n,m,0}$, except that v is replaced by any neighbour of the corner.

This Step is illustrated on Figure 5.

2. (b)

Verification that images of $\phi_{n,m}$ are minimal total dominating sets.

Consider a pattern $p$ of $X^{MT}$ on $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$. The set $\phi_{n,m}(p)$ is a minimal total dominating set of $G_{n,m}$:

• •

Any vertex of $G_{n,m}$ is dominated in $\phi_{n,m}(p)$.

Any (dominant or not) position which was dominated before applying Rule i is still dominated afterwards: indeed, if some position u lies in the neighbourhood of a dominant position v suppressed by Rule i, then since v had no private neighbours in $G_{n,m}$, u is dominated by another position. For similar reasons, no positions become undominated after the application of Rule ii: only the neighbours of some w’ could be affected and if w’ becomes non-dominant it means that they were dominated by other positions, so that they stay dominated. Since all the positions inside $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$ were dominated before applying the rules, it only remains to show that the positions inside $\mathbb{B}_{n,m,0}$ are dominated after applying Rule ii. This is true thanks to this rule: any undominated position u inside the border sees its neighbour v inside $\llbracket 2,n-1\rrbracket\times\llbracket 2,m-1\rrbracket$ becomes dominant. The same works for the corners, except that the neighbour comes from the border.

• •

Any dominant position has a private neighbour.

At the end of Step i, any dominant position has a private neighbour. Only the creation of a domination position v during the execution of Rule ii on position u could affect this property by disabling the private neighbour of a position w in its neighbourhood or by not having any private neighbour itself. The first case cannot happen since any dominant position w’ whose unique private neighbour is w is suppressed. The second one also never happens since the position u is a private neighbour for v. 3. (c)

For all $n,m$, the number of preimages of $\phi_{n,m}$ for any minimal dominating set of $G_{n,m}$ is bounded (roughly) by $2^{4*(2m+2n)}$, since any symbol modified by the application is at distance at most $4$ of the border of $\llbracket 1,n\rrbracket\times\llbracket 1,m\rrbracket$. As a consequence, $N_{n,m}(X^{MT})\leq 2^{8(m+n)}T_{n,m}$.

∎

Theorem 1** (Asymptotic behaviour).**

There exists some $\nu_{D}\geq 0$ (resp. $\nu_{M}$, $\nu_{T}$ and $\nu_{MT}$) such that

[TABLE]

(resp. $M_{n,m}=\nu_{M}^{nm+o(nm)}$, $T_{n,m}=\nu_{T}^{nm+o(nm)}$ and $MT_{n,m}=\nu_{MT}^{nm+o(nm)}$).

Proof.

Let us prove this for the sequence $(M_{n,m})$ (the proof is similar for the other sequences).

As a consequence of Lemma 6, for all $n,m$,

[TABLE]

As a consequence,

[TABLE]

This means that $M_{n,m}=2^{h(X^{M})\cdot nm+o(nm)}=\nu_{M}^{nm+o(nm)}$, where $\nu_{M}=2^{h(X^{M})}$.

∎

4 Computability of the growth rate

In this section, we prove that the growth rate $\nu_{D}$ (resp. $\nu_{M}$, $\nu_{T}$ and $\nu_{MT}$) is a computable number, meaning that there exists an algorithm which computes approximations of this number with arbitrary given precision. For this purpose, we rely on the block-gluing property, defined in Section 4.1, and proved for $X^{D}$ (resp. $X^{M}$, $X^{T}$ and $X^{MT}$) in Section 4.2. If a subshift of finite has this property then its entropy is computable. We describe a known algorithm to compute it.

4.1

The block-gluing property

4.1.1 Definition

For two finite subsets $\mathbb{U},\mathbb{V}$ of $\mathbb{Z}^{2}$, we write

[TABLE]

The usual definition of the block-gluing property is the following one:

Definition 12**.**

For a fixed integer $c\geq 0$, we say that a bidimensional subshift of finite type $X$ on alphabet $\mathcal{A}$ is $c$-block-gluing when for every $n\geq 0$ and any two globally admissible patterns $p$ and $q$ of $X$ on support $\llbracket 1,n\rrbracket^{2}$, for all $\textbf{u},\textbf{v}\in\mathbb{Z}^{2}$ such that $\delta(\textbf{u}+\llbracket 1,n\rrbracket^{2},\textbf{v}+\llbracket 1,n\rrbracket^{2})\geq c$, there exists a configuration $x$ of $X$ such that $x_{\textbf{u}+\llbracket 1,n\rrbracket^{2}}=p$ and $x_{\textbf{v}+\llbracket 1,n\rrbracket^{2}}=q$.

Informally, this means that any pair of rectangular patterns placed at whatever positions can be completed into a configuration of $X$, provided that the distance between the two patterns is at least $c$.

Notation 6**.**

For any subshift of finite type $X$, we denote by $c(X)$ the smallest $c$ such that $X$ is $c$-block-gluing. If $X$ is not block gluing for any integer $c$, we write $c(X)=+\infty$.

In the following, we will use the notations $\mathbb{Z}_{-}=\,\rrbracket-\infty,0\rrbracket$ and $\mathbb{Z}_{+}=\,\rrbracket 0,+\infty\llbracket$. Here is a characterisation of the block-gluing property:

Proposition 3**.**

Let $c\geq 0$ be an integer. A bidimensional subshift $X$ is $c$-block-gluing if and only if for all $k\geq c$ and $p$ and $q$ globally admissible patterns on supports $\mathbb{Z}_{-}\times\mathbb{Z}$ (resp. $\mathbb{Z}\times\mathbb{Z}_{-}$) and $\mathbb{Z}_{+}\times\mathbb{Z}$ (resp. $\mathbb{Z}\times\mathbb{Z}_{+}$), there exists a configuration $x$ in $X$ such that $x_{|\mathbb{Z}_{-}\times\mathbb{Z}}=p$ and $x_{|(k,0)+\mathbb{Z}_{+}\times\mathbb{Z}}=q$ (resp. $x_{|\mathbb{Z}\times\mathbb{Z}_{-}}=p$ and $x_{|(k,0)+\mathbb{Z}\times\mathbb{Z}_{+}}=q$).

Informally, in order to check the block-gluing property, it is sufficient to prove that any two patterns on half-planes can be glued with arbitrary distance greater than $c$ in a configuration of $X$.

Proof.

• •

$(\Leftarrow)$: Let us assume that $X$ verifies the second hypothesis. Let us consider some integer $n$, and two globally admissible patterns $\overline{p},\overline{q}$ of $X$ on support $\llbracket 1,n\rrbracket^{2}$. Let $\textbf{u},\textbf{v}$ be two positions such that $\delta(\textbf{u}+\llbracket 1,n\rrbracket^{2},\textbf{v}+\llbracket 1,n\rrbracket^{2})\geq c$. This means that the two translates $\textbf{u}+\llbracket 1,n\rrbracket^{2}$ and $\textbf{v}+\llbracket 1,n\rrbracket^{2}$ have more than $c$ columns separating them or more than $c$ rows. Without loss of generality, we assume that we are in the case of separating columns, and denote by $k\geq c$ the exact number of columns separating $\overline{p}$ and $\overline{q}$. Since $\overline{p}$ and $\overline{q}$ are globally admissible, there exist $p$ and $q$ globally admissible patterns of $X$ on respective supports $\mathbb{Z}_{-}\times\mathbb{Z}$ and $(k,0)+\mathbb{Z}_{+}\times\mathbb{Z}$ whose restrictions on $-(n,0)+\llbracket 1,n\rrbracket^{2}$ and $\textbf{v}-\textbf{u}-(n,0)+\llbracket 1,n\rrbracket^{2}$ are respectively $\overline{p}$ and $\overline{q}$. By hypothesis, there exists some configuration $x$ of $X$ whose restrictions on $\mathbb{Z}_{-}\times\mathbb{Z}$ and $(k,0)+\mathbb{Z}_{+}\times\mathbb{Z}$ are respectively $p$ and $q$. The patterns $\overline{p}$ and $\overline{q}$ can be found on $\textbf{u}+\llbracket 1,n\rrbracket^{2}$ and $\textbf{v}+\llbracket 1,n\rrbracket^{2}$ in the configuration $\sigma^{\textbf{u}}x$.

• •

$(\Rightarrow)$: Let us assume the first hypothesis on $X$ is true, and let $p$ and $q$ be two patterns on supports $\mathbb{Z}_{-}\times\mathbb{Z}$ and $(k,0)+\mathbb{Z}_{+}\times\mathbb{Z}$ for some $k\geq c$ (the other case is proved in a similar way). From the block-gluing property, for all $n\geq 0$ one can extend the restriction of $p$ on $\llbracket 1,n\rrbracket^{2}-(n,0)$ and the restriction of $q$ on $\llbracket 1,n\rrbracket^{2}-(n,0)+(k+1,0)$ into a configuration $x_{n}\in X$. By compactness of the set $X$ for the product of the discrete topology, this sequence admits a subsequence which converges to some $x\in X$. This $x$ verifies $x_{|\mathbb{Z}_{-}\times\mathbb{Z}}=p$ and $x_{|(k,0)+\mathbb{Z}_{+}\times\mathbb{Z}}=q$.

∎

4.1.2 Algorithmic computability of

entropy

Definition 13**.**

Let $f:\mathbb{N}\rightarrow\mathbb{N}$ a computable function. A real number $x$ is said to be computable with rate $f$ when there exists an algorithm which, given an integer $n$ as input, outputs in at most $f(n)$ steps a rational number $r_{n}$ such that $|x-r_{n}|\leq\frac{1}{n}$.

This definition corresponds to Definition 1.3 in [Pavlov Schraudner]. The following theorem is Theorem 1.4 in the same reference. Its proof provides an algorithm to compute $h(X)$.

Theorem 2** ([Pavlov Schraudner]).**

Let $X$ be a block-gluing bidimensional subshift of finite type. Then $h(X)$ is computable with rate $n\mapsto 2^{O(n^{2})}$.

Lemma 8**.**

Let $X$ be a $c$-block gluing bidimensional subshift of finite type on alphabet $\mathcal{A}$. For all $k\geq 1$, the number $N_{k}(X)$ is equal to the number of $k\times k$ patterns which appear in a $\left(|\mathcal{A}|^{2c+1}\cdot(c+k)+1\right)\times(2c+k+2)$ locally-admissible rectangular pattern whose restrictions on the two extremal vertical (resp. horizontal) sides are equal.

Remark 1**.**

Let us note that in general the entropy of a bidimensional subshift of finite type is not computable at all ([Hochman Meyerovitch] Theorem 1.1 and the existence of non-computable right recursively enumerable numbers).

This algorithm is as follows:

4.2 Proof of the

block-gluing property for domination subshifts

It is straightforward to check that the domination subshift $X^{D}$ and the total domination subshift $X^{T}$ satisfy the block-gluing property, with $c(X^{D})=1$ (just fill every cell with grey). In this section, we prove that $X^{M}$ and $X^{MT}$ also satisfy this property.

Notation 7**.**

In the following, for all $j\in\mathbb{Z}$, we denote by $C_{j}$ the column $\{j\}\times\mathbb{Z}$ of $\mathbb{Z}^{2}$.

Theorem 3**.**

The minimal domination subshift is block gluing and $c(X^{M})=5$.

Idea of the proof: In order to simplify the proof of the block-gluing property, we rely on Proposition 3. The proof of the block-gluing property for two half-plane patterns consists in determining successively the intermediate columns from the patterns towards the ”center” (chosen to be column $C_{k-2}$, for concision). The completion follows an algorithm which ensures that, when the number of intermediate columns is great enough, any added dominant element has a private neighbour in an already constructed column or is isolated. This ensures that the rules of the subshift are not broken.

Proof.

• •

Filling the intermediate columns between two half-plane patterns.

Let $p$ and $q$ be two patterns respectively on $\mathbb{Z}_{-}\times\mathbb{Z}$ and $\mathbb{Z}_{+}\times\mathbb{Z}$ (the proof for the other case is similar). Let us determine a configuration of $\mathcal{A}_{0}^{\mathbb{Z}^{2}}$ such that $x_{|\mathbb{Z}_{-}\times\mathbb{Z}}=p$ and $x_{|(k,0)+\mathbb{Z}_{+}\times\mathbb{Z}}=q$. The intermediate columns $C_{1},...,C_{k}$ are determined by the following algorithm:

Filling the intermediate columns, from $C_{1}$ to $C_{k-3}$, then $C_{k},C_{k-1}$.

Successively, for all $j$ from $1$ to $k-3$, we determine the column $C_{j}$ according to the following rule: for all $\textbf{u}\in C_{j}$, $x_{\textbf{u}}$ is when $x_{\textbf{u}-(1,0)}$, $x_{\textbf{u}-(1,1)}$, $x_{\textbf{u}-(1,-1)}$ and $x_{\textbf{u}-(2,0)}$ are (see Figure 6). Else, $x_{\textbf{u}}$ is set to . Similarly, for $j=k$ and then $j=k-1$, we determine $x$ on any position $x_{\textbf{u}}$ for $\textbf{u}\in C_{j}$ by applying a symmetrical rule: $x_{\textbf{u}}$ is when $x_{\textbf{u}+(1,0)}$, $x_{\textbf{u}+(1,1)}$, $x_{\textbf{u}+(1,-1)}$ and $x_{\textbf{u}+(2,0)}$ are . Else $x_{\textbf{u}}$ is set to . 2. 2.

The central column $C_{k-2}$.

We now determine $x$ on the central column $C_{k-2}$. For all $\textbf{u}\in C_{k-2}$, $x_{\textbf{u}}$ is when $x_{\textbf{u}+(1,0)}$, $x_{\textbf{u}+(1,1)}$, $x_{\textbf{u}+(1,-1)}$ and $x_{\textbf{u}+(2,0)}$ are equal to , or when $x_{\textbf{u}-(1,0)}$, $x_{\textbf{u}-(1,1)}$, $x_{\textbf{u}-(1,-1)}$ and $x_{\textbf{u}-(2,0)}$ are equal to . Else it is . 3. 3.

Eliminating non-domination errors in the central column. Choose any position $\textbf{u}_{0}\in C_{k-2}$ and check if this position has a symbol in its neighbourhood. If not, then set the symbol on this position. Repeat this from $\textbf{u}_{0}+(0,1)$ upwards, and in parallel from $\textbf{u}_{0}-(0,1)$ downwards.

See an illustration of this algorithm on Figure 7.

• •

The obtained configuration is in $X^{M}$.

We have to check that the configuration $x$ we constructed satisfies the local rules of the minimal domination subshift.

Local rules are verified inside the half-planes.

By hypothesis, the patterns $p$ and $q$ are globally admissible in $X^{M}$. As a consequence, for all u in $\rrbracket-\infty,-2\rrbracket\times\mathbb{Z}$ or $\llbracket k+3,+\infty\llbracket\times\mathbb{Z}$, $x_{|\textbf{u}+\llbracket-2,2\rrbracket^{2}}$ is not a forbidden pattern. We have left to check that no forbidden patterns are created through the execution of the algorithm described in the first point of the proof. 2. 2.

Every position outside $\text{supp}(p)\cup\text{supp}(q)$ and not in $S$ is dominated.

In the columns $C_{-1}$ and $C_{k+2}$, this comes from the fact that the patterns $p$ and $q$ are globally admissible. For $j$ between [math] and $k-3$, and $\textbf{u}\in C_{j}$, if u is not dominated by a position in $C_{j}$ or $C_{j-1}$ the position $\textbf{u}+(1,0)$ is the symbol (by the first and second steps of the algorithm), and thus u is dominated. A symmetrical reasoning works for the positions in the columns $C_{k+1},C_{k},C_{k-1}$. For a position in the central column $C_{k-2}$, this is guaranteed by Step 3. 3. 3.

Every dominant position outside $\text{supp}(p)\cup\text{supp}(q)$ is isolated or has a private neighbour not in $S$.

Let us consider a non-isolated dominant position u.

1. (a)

If it lies in $C_{-1}$ (resp. $C_{k+2}$), u has a private neighbour in a configuration of $X$ that extends $p$ (resp. $q$). If this private neighbour is in column $C_{-2}$ or $C_{-1}$ (resp. $C_{k+2}$ or $C_{k+3}$), then it stays a private neighbour of u in $x$. If it is $\textbf{u}+(1,0)$ (resp. $\textbf{u}-(1,0)$), then it stays a private neighbour in $x$: since this position is dominated by u according to the first step of the algorithm (resp. second step), it is not dominated in $x$ by a position in $C_{0}$ (resp. $C_{k+1}$). The same reasoning is applied to positions in columns $C_{0}$ and $C_{k+1}$. 2. (b)

In the other columns $C_{j}$ for $j<k-2$, the first step guarantees, for any position u in $C_{j}$ that is a dominant position, that the position $\textbf{u}-(1,0)$ is a private neighbour. A similar reasoning applies to column $C_{k}$ and $C_{k-1}$. 3. (c)

If u is in the column $C_{k-2}$, it means that it was introduced in either the second or the third step, meaning that it has a private neighbour in column $C_{k-3}$ or $C_{k-1}$.

• •

The subshift $X^{M}$ is not $4$-block-gluing.

We consider the two half-plane patterns $p$ and $q$ on respective supports $\mathbb{Z}_{-}\times\mathbb{Z}$ and $\mathbb{Z}_{+}\times\mathbb{Z}$ such that for all $j\leq 0$, if $-j\equiv 0,1[4]$, then for all $\textbf{u}\in C_{j}$, $p_{\textbf{u}}$ is , else for all $\textbf{u}\in C_{j}$, it is , and $q$ is obtained from $p$ by symmetry. It is easy to see that these patterns are globally admissible. We leave 4 columns between $p$ and $q$ (see Figure 8) To ensure that the dominant positions in columns 0 and 5, which are not isolated, have private neighbours, every cell of the four middle columns needs to be , as in Figure 8. This filling implies that the cells in column 2 and 3, which are not dominant, are also not dominated. This shows that the subshift is not 4 block gluing.

As a consequence $c(X^{M})>4$. Since it is $5$-block-gluing, $c(X^{M})=5$.

∎

Theorem 4**.**

The minimal total domination subshift is block gluing and $c(X^{MT})=5$.

Idea of the proof: We follow the same scheme as in the proof of Theorem 3, except that we have to take into account the variations in the definition of the subshift $X^{MT}$. For the sake of readability, we reproduce the structure of the proof.

Proof.

• •

Filling the intermediate columns between two half-plane patterns.

We provide here an algorithm to fill these columns between two patterns $p$ and $q$ respectively on $\mathbb{Z}_{-}\times\mathbb{Z}$ and $\mathbb{Z}_{+}\times\mathbb{Z}$ into a configuration $x\in X^{MT}$:

Filling the intermediate columns, from $C_{1}$ to $C_{k-3}$, then $C_{k},C_{k-1}$. Successively, for all $j$ from $1$ to $k-3$, we determine the column $C_{j}$ according to the following rule: for all $\textbf{u}\in C_{j}$, $x_{\textbf{u}}$ is when $x_{\textbf{u}-(1,1)}$, $x_{\textbf{u}-(1,-1)}$ and $x_{\textbf{u}-(2,0)}$ are (the difference with the proof of Theorem 3 is that the symbol $x_{\textbf{u}-(1,0)}$ is not imposed). Else, $x_{\textbf{u}}$ is set to . This rule is illustrated in Figure 10:

For $j=k$ and then $j=(k-1)$, we determine $x$ on any position $x_{\textbf{u}}$ for $\textbf{u}\in C_{j}$ by applying a symmetrical rule: $x_{\textbf{u}}$ is when $x_{\textbf{u}+(1,1)}$, $x_{\textbf{u}+(1,-1)}$ and $x_{\textbf{u}+(2,0)}$ are . Else it is . 2. 2.

The central column ($j=k-2)$.

We then determine $x$ on the central column $C_{k-2}$. For all $\textbf{u}\in C_{k-2}$, $x_{\textbf{u}}$ is when $x_{\textbf{u}+(1,1)}$, $x_{\textbf{u}+(1,-1)}$ and $x_{\textbf{u}+(2,0)}$ are equal to , or when $x_{\textbf{u}-(1,1)}$, $x_{\textbf{u}-(1,-1)}$ and $x_{\textbf{u}-(2,0)}$ are equal to . Else it is . 3. 3.

Eliminating minimality and total domination errors in the central column.

Choose any position $\textbf{u}_{0}\in C_{k-2}$. From this position upwards, check for all positions if they are dominated. If this is not the case, then change the symbol on $\textbf{u}+(0,1)$ into . After $\textbf{u}_{0}$ has been processed do the same symmetrically (change the symbol in $\textbf{u}_{0}-(0,1)$ when u is not dominated) in parallel downwards, beginning from $\textbf{u}_{0}-(0,1)$.

See an illustration of this algorithm on Figure 7.

• •

The obtained configuration is in $X^{MT}$.

We have to check that the local rules of the minimal total domination subshift are verified over all the constructed configuration $x$.

Local rules are verified inside the half-planes.

Same as the corresponding point in the proof of Theorem 3. 2. 2.

Every position outside $\text{supp}(p)\cup\text{supp}(q)$ is dominated.

Same as the corresponding point in the proof of Theorem 3 for the positions outside the central column $C_{k-2}$. In this column, let us assume that a position u above $\vec{u}_{0}$ (without loss of generality) is not dominated. Then the last step of the algorithm, when scanning this position, would have changed the symbol on position $\textbf{u}+(0,1)$, which is a contradiction.

In particular, no dominant positions are isolated. 3. 3.

Every dominant position outside $\text{supp}(p)\cup\text{supp}(q)$ has a private neighbour.

1. (a+b)

Outside the central column, the proof is similar to the corresponding points in the proof of Theorem 3. 2. (c)

In the column $C_{k-2}$, the dominant positions added in Step 2 necessarily have a private neighbour in column $C_{k-3}$ or $C_{k-1}$. Let us take a dominant position u, assumed without loss of generality to be above $\textbf{u}_{0}$, which was added in the last step. This implies that $\textbf{u}-(0,1)$ was not dominated when the algorithm checked this position. As a consequence, it is a private neighbour for u.

• •

The subshift $X^{MT}$ is not $4$-block-gluing.

Let us consider the patterns $p$ and $q$ defined in the corresponding point in the proof of Theorem 3. It is easy to see that these two patterns are also globally admissible in the subshift $X^{MT}$. We only have to check that in the constructed configuration, no dominant positions are isolated, which is straightforward.

Using the same arguments as the ones for the minimal domination case, it is easy to see that any configuration in $\mathcal{A}^{\mathbb{Z}^{2}}$ where $p$ and $q$ are glued at distance 4 contains some forbidden patterns.

As a consequence $c(X^{MT})>4$. Since it is $5$-block-gluing, $c(X^{MT})=5$.

∎

As a direct consequence of Theorem 2:

Theorem 5**.**

The numbers $\nu_{D}$ and $\nu_{T}$ are computable with rate $n\mapsto 2^{n^{2}}$. The numbers $\nu_{M}$ and $\nu_{MT}$ are computable with rate $n\mapsto 2^{5n^{2}}$.

5 Computing bounds for the growth rate

Although the algorithm presented in Section 4.1.2 provides a way to compute the growth rates of various dominating sets of the grids $G_{n,m}$, it is not efficient enough for practical use on a computer. In this section, we use other tools which make it possible to obtain bounds for the growth rates, although with no guarantee on their precision. These bounds are obtained using computer resources, by running a C++ program made for the occasion. The technique relies on, for a fixed $m$, assimilating the dominating sets of $G_{n,m}$ to patterns of a unidimensional subshift of finite type, whose entropy is known to be computable through linear algebra computing.

This method is well known. It was for instance used, along with other techniques, to solve the problem of finding the minimum size of a dominating (see [Gonçalves et al.]), 2-dominating and Roman dominating (see [Rao Talon]) set of a grid of arbitrary size. These papers provide an alternate explanation without relating it to the theory of SFTs. For instance, Section 2 in [Rao Talon] uses the same technique as the one we use here, but in the (min,+)-algebra. However, in their paper there are no such things as lower or upper bounds we investigate here: they only enumerate sets which are exactly 2-dominating or Roman dominating. Since they are interested in finding the minimum size of such a set, they can apply some optimisations to avoid enumerating some sets which cannot be of minimum size. Since we want to count all the different dominating sets, these optimisations do not apply here.

5.1 Relating $(D_{n,m})$ to the entropy of some unidimensional SFT

5.1.1 Nearest-neighbour unidimensional subshifts of finite type

In this section $\mathcal{A}=(a_{1},...,a_{k})$ is a finite set, and $X$ a unidimensional subshift of finite type on alphabet $\mathcal{A}$. Let us denote by $(e_{1},...,e_{k})$ the canonical basis of $\mathbb{R}^{k}$.

Definition 14**.**

The subshift $X$ is said to be nearest neighbour when it is defined by forbidding a set of patterns on support $\{0,1\}$.

Definition 15**.**

The adjacency matrix of $X$ is the matrix $M\in\mathcal{M}_{k}(\mathbb{R})$ such that $M[e_{i},e_{j}]=1$ if the pattern $a_{i}a_{j}$ is not forbidden, or 0 otherwise.

The following is well known:

Proposition 4**.**

Let $||.||$ be any matricial norm. The entropy of $X$ is equal to the spectral radius of $M$:

[TABLE]

5.1.2 Unidimensional versions of the domination subshifts

We define here the unidimensional versions of the domination subshifts defined in Section 3.2. We use them to describe and prove the method we use to obtain the bounds on the growth rates. The first one ($X^{D,m}$) is used to obtain the lower bound, whereas we use the second one ($X_{*}^{D,m}$) for the upper bound.

Notation 8**.**

Let us fix some integer $m\geq 1$. We denote by $X^{D,m}$ the undimensional subshift on alphabet $\mathcal{A}_{0}^{n}$ such that a configuration $x$ is in $X^{D,m}$ if and only if the set of positions $(j,k)\in\mathbb{Z}\times\llbracket 1,m\rrbracket$ such that the symbol $(x_{j})_{k}$ is grey forms a dominating set of the grid $\mathbb{Z}\times\llbracket 1,m\rrbracket$.

With similar arguments as in the proofs of Lemma 6 and Lemma 7, we get that when $m$ is fixed and $n$ grows to infinity:

[TABLE]

Notation 9**.**

For all $m\geq 3$, we also denote by $X_{*}^{D,m}$ the undimensional subshift on alphabet $\mathcal{A}_{0}^{n}$ such that a configuration $x$ is in $X_{*}^{D,m}$ if and only if the set of positions $(j,k)\in\mathbb{Z}\times\llbracket 2,m-1\rrbracket$ such that the symbol $(x_{j})_{k}$ is grey forms a dominating set of the grid $\mathbb{Z}\times\llbracket 2,m-1\rrbracket$.

5.1.3 Recoding into nearest-neighbour

subshifts

Let us set $\mathcal{A}_{1}=\left\{\leavevmode\hbox to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@stroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{0.85,0.85,0.85}\pgfsys@color@gray@fill{0.85}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}},\leavevmode\hbox to8.94pt{\vbox to8.94pt{\pgfpicture\makeatletter\hbox{\hskip 0.2pt\lower-0.2pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@rgb@stroke{0}{0}{0}\pgfsys@invoke{ }\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{ }\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{ }{{}} {}{{}}{} {}{{}}{}{}{}{}{{}}{}\pgfsys@beginscope\pgfsys@invoke{ }\definecolor{pgffillcolor}{rgb}{0.575,0.575,0.575}\pgfsys@color@gray@fill{0.575}\pgfsys@invoke{ }{}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@moveto{0.0pt}{0.0pt}\pgfsys@lineto{0.0pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{8.5359pt}\pgfsys@lineto{8.5359pt}{0.0pt}\pgfsys@closepath\pgfsys@moveto{8.5359pt}{8.5359pt}\pgfsys@fillstroke\pgfsys@invoke{ } \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}\right\}$, and let us consider the map $\varphi:\left(\mathcal{A}^{m}_{0}\right)^{\mathbb{Z}}\rightarrow\left(\mathcal{A}^{m}_{1}\right)^{\mathbb{Z}}$ that acts on configurations of $\left(\mathcal{A}^{n}_{0}\right)^{\mathbb{Z}}$ by changing the $i^{\text{th}}$ symbol of any position $j\in\mathbb{Z}$ into whenever it is not dominant and dominated by an element of $C_{j-1}\bigcap\left(\mathbb{Z}\times\llbracket 1,m\rrbracket\right)$ or $C_{j}\bigcap\left(\mathbb{Z}\times\llbracket 1,m\rrbracket\right)$. Informally, from lightest to darkest they stand for an undominated cell (which is not dominant), a dominated cell which is not dominant and a dominant cell. This is illustrated in Figure 11. The nearest-neighbour property makes it possible to count the dominating sets without enumerating them fully: it is enough to store a small number of the latest columns, proceeding from left to right in the grid.

Since $\varphi$ commutes with the shift action and is invertible,

[TABLE]

Moreover, the subshift $X^{D,m}$ has the nearest-neighbour property.

5.2 Numerical approximations

We use these last equalities to prove the following:

Theorem 6** (domination).**

The following inequalities hold $1.950022198\leq\nu_{D}\leq 1.959201684$.

Proof.

• •

Lower bound:

For all integers $n,m_{1},m_{2}$, $D_{n,m_{1}+m_{2}}\geq D_{n,m_{1}}\cdot D_{n,m_{2}}.$

Indeed, let us consider two sets dominating respectively $G_{n,m_{1}}$ and $G_{n,m_{2}}$. By gluing the first one on the top of the second one, we obtain a dominating set of $G_{n,m_{1}+m_{2}}$. This is true because any position in this grid is either in the copy of the grid $G_{n,m_{1}}$ and thus dominated by an element in this grid, or in the copy of $G_{n,m_{2}}$. Since this construction is invertible, we obtain the announced inequality. 2. 2.

As a consequence, for all $k\geq 0$,

[TABLE]

where the function $o(n)$ is related to the fact that we used $18$ lines. This implies that

[TABLE] 3. 3.

This number is equal to $h(\varphi(X^{D,18}))$, which is computed using Section 5.1.1. The lower bound follows.

• •

Upper bound:

For all $n,m$, let us denote by $D^{*}_{n,m}$ the number of sets of vertices of $G_{n,m}$ which dominate the middle $m-2$ lines (i.e. cells of the first and last lines might not be dominated). We have a direct inequality

[TABLE] 2. 2.

For a reason similar as the one in the first point of the proof of the lower bound, for all $m_{1},m_{2}$, $D^{*}_{n,m_{1}+m_{2}}\leq D^{*}_{n,m_{1}}\cdot D^{*}_{n,m_{2}}$. 3. 3.

For all $k\geq 0$ and $n\geq 0$,

[TABLE]

As a consequence

[TABLE]

With the same method as for the lower bound, we obtain the upper bound.

∎

Remark 2**.**

With further numerical manipulations, we notice that the lower bound and the upper bound seem to get closer to each other rather slowly. To speed up the convergence, we had the idea of using the sequences of ratios $h(X^{D,m+1})/h(X^{D,m})$ and $h(X_{*}^{D,m+1})/h(X_{*}^{D,m})$. This seems to offer a much better convergence speed. Indeed, for both sequences, from $m=11$ on, the ratio seem to be stabilised around $1.954751195$.

Using the same method, we provide bounds for some other problems. For the total domination, we can make the computations until $m=17$. However, for the other problems (the ones with the minimality constraint) the number of patterns we enumerate grows exponentially at a much faster rate than for the domination problem, thus the bounds are less good. We cannot go further than around $m=10$ for these problems.

Theorem 7** (total domination).**

$1.904220376\leq\nu_{T}\leq 1.923434191$.

Remark 3**.**

As in Remark 2, the ratios $h(X^{T,m+1})/h(X^{T,m})$ and $h(X_{*}^{T,m+1})/h(X_{*}^{T,m})$) offer a much better convergence speed. Indeed, from $m=10$ they seem to stabilise, both around $1.915316$.

Theorem 8** (minimal domination).**

$1.315870482\leq\nu_{M}\leq 1.550332154$.

Theorem 9** (minimal total domination).**

$1.275805204\leq\nu_{MT}\leq 1.524476040$.