Asymptotic growth rate of square grids dominating sets: A symbolic dynamics approach
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 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 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 [1], proving Chang's conjecture, which tells that denoting the finite rectangular grid, 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 [2].
In the present text, we are interested in the asymptotic growth rate of the number of dominating sets, on the finite rectangular grids , 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:
- 1.
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.
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 . Comparing the number of dominating sets on finite grids and the number of patterns which appear in configurations of the corresponding subshift, we prove existence of a growth rate and show that it is equal to the entropy of the dynamical system.
- 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.
In Section 5, we provide some bounds for the growth rates obtained by a computer program.
Section snippets
Definitions
In the following, for a graph , we will say that two vertices in V are neighbours or connected when the edge is in E. For all , we will denote by the finite square grid graph of size .
Definition 1 Let be a graph. A subset is said to be a dominating set of the graph G when every has a neighbour in S. It is said to be a minimal dominating set of G when it is a dominating set of G and for all , is not dominating. It is said total dominating when for all v
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.
Computability of the growth rate
In this section, we prove that the growth rate (resp. , and ) 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 (resp. , and ) in Section 4.2. If a subshift of finite type has this property then its entropy is computable. We describe a known algorithm to compute it.
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 , 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
Declaration of Competing Interest
We wish to confirm that there are no known conflicts of interest associated with this publication and there has been no significant financial support for this work that could have influenced its outcome.
Acknowledgements
This project was supported by the ANR project ANR-16-CE40-0005.
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