A dominating set of a graph is a subset of its vertices such that any vertex not in is connected to a vertex in . 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 and the finite graph 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 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 [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 , when and 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.
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 the existence of a growth rate and show that it is equal to the entropy of the dynamical system.
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.
In Section 5, we provide some bounds for the growth rates obtained by a computer program.
2 Notions of dominating sets of square grids
In the following, for a graph , we will say that two vertices in are neighbours or connected when the edge is in . For all , we will denote by the finite square grid graph of size .
Let be a graph, a subset of and . We say that is dominated by if has a neighbour in .
Let be a graph. A subset is said to be a dominating set of the graph when every is dominated by . It is said to be a minimal dominating set of when it is a dominating set of and for all , is not dominating. It is said total dominating when for all in , has a neighbour in .
A subset is said to be minimal total dominating when it is total dominating and for all , is not total dominating.
It is straightforward to see that a total-minimal-dominating set is also minimal total dominating, but the converse is not true: see for instance Figure 1. Notice that a minimal dominating set (resp. minimal total dominating set) is a dominating set (resp. total dominating set) which is inclusion-wise minimal.
When a dominating set of a graph is fixed, a vertex is called a dominant element of when it is in , and a dominated element when it has a neighbour in . A neighbour of a dominant element is said to be a private neighbour of when is the only neighbour of in the set .
These notions are illustrated in Figure 1.
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 is dominating (or minimal dominating, etc) by checking, for each vertex, whether or not this vertex and its neighbours are in
Let be a set of vertices of a graph . Then for all and such that is not a neighbour of , is dominated by if and only if it is dominated by .
Let be a set of vertices of a graph . We say that a dominant element is isolated in when it has no neighbours in .
Let be a dominating set of a graph . is minimal dominating if and only if any of its elements is isolated in or has a private neighbour not in .
: Let us assume that is minimal dominating, and fix . From Fact 1 and by definition of a minimal dominating set, any which is not in the neighbourhood of is dominated by . Since is not dominating, it means that:
is not dominated by , which means that is isolated in ,
or there exists some neighbour of which is not dominated by , hence is a private neighbour of which is not in .
: Conversely, let us fix some dominating set such that every has a private neighbour not in or is isolated. Fix some . If it has a private neighbour , then is not dominated by , and thus is not dominating. If it has no private neighbours, then it is isolated. This means that is not dominated by , therefore the set is not dominating. In both cases, we conclude that is minimal dominating.
With a similar proof, we obtain the following:
A total dominating set of a graph is minimal total dominating if and only if any has a private neighbour and is not isolated.
In the following, we will use the following notations:
In the following, for all integers , we denote by , and respectively the number of dominating sets of the grid , 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
Let be a finite set, and integer. A pattern on alphabet is an element of for some finite . The set is called the support of , and is denoted . Informally, it is the location of in the grid .
For a configuration of (resp. a pattern for some ), we denote by the restriction of to some subset (resp. the restriction of to ).
Let be a finite set, and integer. A -dimensional subshift of finite type (SFT) on alphabet is a subset of defined by a finite set of forbidden patterns. Formally, a subset of is a subshift of finite type when there exists some finite sets and such that:
The elements of 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.
Let us denote by the -shift action on defined such that for all ,
Informally, acts on a configuration by translating it by the vector
acts on a configuration by translating it by the vectoru.
For a SFT a globally admissible pattern of size is some which appears in a configuration of , that is when . When , we extend the definition to patterns when there exists a configuration of such that .
For a subshift of finite type , we denote by . the number of globally admissible patterns of size . When , we extend the notation and denote by resp. the number of globally admissible pattenrs of size .
The topological entropy of a subshift of finite type is the number
The following three lemmas are well known (see for instance [Lind Marcus]).
The infimum in the definition of is in fact a limit:
A conjugation between two -dimensional subshifts of finite type and is an invertible map such that for all and , . In this case, and are said to be conjugated.
If two subshifts of finite type and are conjugated, then .
Let be a bidimensional subshift of finite type. Then:
3.2 Domination subshifts
In this section, the alphabet is , and .
The domination (resp. minimal domination, total domination and minimal total-domination) denoted by (resp. , and ), is the set of elements of such that is a dominating (resp. minimal dominating, total dominating and minimal total dominating) set of the infinite square grid . In all these cases, a configuration of the subshift is called a dominated configuration. We also say that u is a dominant position of the configuration when is grey. Likewise, a private neighbour is still a position which is dominated by exactly one dominant position.
The sets , , and are subshifts of finite type.
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 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 for (resp. , and ) with the number of dominating sets (resp. minimal dominating sets, total dominating set and minimal total dominating sets) of . 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 to any translate of and assimilate any set of vertices of a finite grid with the pattern of on defined by being grey if and only if .
If is a subset of , we define the (extended) neighbourhood of as
We also define, for all and the border
For convenience, we extend the notation to .
For all , the following inequalities hold: .
For all , any dominating set of can be extended into a configuration of by defining the symbol of any position outside to be grey. As a consequence, any dominating set of is globally admissible in and thus .
Any pattern of on can be turned into a dominating set of by extending it with grey symbols. Hence we obtain the inequality for all .
For all , the following inequalities hold:
A completion algorithm of a dominating set into a configuration of . Let be a minimal dominating set of . Let us extend it into a configuration of using the following algorithm: successively for every , we extend the current pattern into a pattern on using the following operations, for all :
if u is a corner then is white;
if u is a neighbour of a corner in one of the vertical sides of then is white;
for all other u, is grey if and only if its neighbour in is neither dominated by an element in this set, nor a dominant element.
This algorithm is illustrated in Figure 2.
Figure 2: Illustration of the completion algorithm in : steps of the algorithm are applied successively from left to right.
The output obtained by repeating the algorithm is a configuration of .
Every position is dominated. This is verified for the positions in . Outside this set, if a position in some (for ) is not dominated before extending the configuration on , 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 , then it has a private neighbour since the pattern on is a minimal dominating set of . Else lies in some for some and there are two cases:
u is not a corner. Its neighbour is white by the application of the algorithm. Also, since its neighbours in are thus dominant or dominated, their neighbours in are white. Moreover, the neighbour of v in 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.
Figure 3: Illustration of the proof of a private neighbour for a non-isolated position. Steps of the completion algorithm for applied from left to right.
Transformating patterns of into minimal dominating sets. Let us define an application which, to each pattern of on , associates a minimal dominating set of defined by:
suppressing any dominant position in which has no private neighbours in and which is dominated by an element of ;
changing successively any non-dominant position of which is still not dominated into a dominant one;
successively, for every dominant position : if one of u’s neighbours v is the only private neighbour of a position w which is not isolated in then change w into a non-dominant position.
This Step is illustrated on Figure 4.
Figure 4: Illustration of the second and then third steps of the algorithm defining for , from left to right. and w are instances of the positions described in Rule iii.
Verifying that images of are minimal dominating sets.
Let us consider a globally admissible pattern of on . The set is a minimal dominating set of :
Any vertex of is dominated or dominant in .
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 , only the positions on the border may not have any private neighbour. After Step i, every
dominant position on 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.
For all , the number of preimages of for any minimal dominating set of is bounded (roughly) by , since any symbol modified by the application is at distance at most 2 from . As a consequence, .
For all , the following bounds hold:
For readability, we reproduce the structure of proof of Lemma 5, but simplify the arguments and refer this proof.
A completion algorithm of dominating set into a configuration of .
Consider a minimal total dominating set of . Any element in is dominated by an element of , and any dominant element in is not isolated and has a private neighbour in (which may or may not be a dominant position). Let us extend it into a configuration of using an algorithm very similar to the one in the corresponding point in the proof of Lemma 5, but the condition in the third point is different:
if u is a corner then is white;
if u is a neighbour of a corner in one of the vertical sides of then is white;
for all other u, is grey if and only if its neighbour in is not dominated by an element in this set.
The result of the algorithm is a configuration of .
Every position is dominated. Similar to the corresponding point in the proof of Lemma 5. 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 , since the pattern on is a minimal total dominating set of , we know that it has a private neighbour. Else, it lies in some for , or in . Then there are two cases:
u is not a corner. If it has no dominant neighbours in Let us call v its neighbour in . Note that, depending on whether or not u is dominated inside , v may be white or grey. Since the neighbours of u in are dominated, v’s neighbours in are white. Finally, since v is dominated by u, its neighbour in is white, hence v is a private neighbour for u.
u is a corner. We apply a similar reasoning.
A transformation of patterns of into dominating sets.
Let us define once again an application which, to each pattern of on , associates a minimal total dominating set of , defined in a similar way as in the corresponding point in the proof of Lemma 5, but the proof is more complex.
suppress any dominant position on the border which has no private neighbours in .
Successively, for every non-corner undominated position u on the border , do the following:
Consider the position v, neighbour of u in . 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 , except that v is replaced by any neighbour of the corner.
This Step is illustrated on Figure 5.
Figure 5: Illustration of the second and then third steps of the algorithm defining for , from left to right. and w’ are instances of the positions described in Rule ii.
Verification that images of are minimal total dominating sets.
Consider a pattern of on . The set is a minimal total dominating set of :
Any vertex of is dominated in .
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 , 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 were dominated before applying the rules, it only remains to show that the positions inside 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 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.
For all , the number of preimages of for any minimal dominating set of is bounded (roughly) by , since any symbol modified by the application is at distance at most of the border of . As a consequence, .
Theorem 1 (Asymptotic behaviour).
There exists some (resp. , and ) such that
(resp. , and ).
Let us prove this for the sequence (the proof is similar for the other sequences).
As a consequence of Lemma 5, for all ,
As a consequence,
This means that , where .
4 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 has this property then its entropy is computable. We describe a kown algorithm to compute it.
4.1 The block-gluing property
For two finite subsets of , we write
The usual definition of the block-gluing property is the following one:
For a fixed integer , we say that a bidimensional subshift of finite type on alphabet is -block-gluing when for every and any two globally admissible patterns and of on support , for all such that , there exists a configuration of such that and .
Informally, this means that any pair of rectangular patterns placed at whatever positions can be completed into a configuration of , provided that the distance between the two patterns is at least .
For any subshift of finite type , we denote by the smallest such that is -block-gluing. If is not block gluing for any integer , we write .
In the following, we will use the notations and . Here is a characterisation of the block-gluing property:
Let be an integer. A bidimensional subshift is -block-gluing if and only if for all and and globally admissible patterns on supports (resp. ) and (resp. ), there exists a configuration in such that and (resp. and ).
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 in a configuration of .
: Let us assume that verifies the second hypothesis. Let us consider some integer , and two globally admissible patterns of on support . Let be two positions such that . This means that the two translates and have more than columns separating them or more than rows. Without loss of generality, we assume that we are in the case of separating columns, and denote by the exact number of columns separating and . Since and are globally admissible, there exist and globally admissible patterns of on respective supports and whose restrictions on and are respectively and . By hypothesis, there exists some configuration of whose restrictions on and are respectively and . The patterns and can be found on and in the configuration .
: Let us assume the first hypothesis on is true, and let and be two patterns on supports and for some (the other case is proved in a similar way). From the block-gluing property, for all one can extend the restriction of on and the restriction of on into a configuration . By compactness of the set for the product of the discrete topology, this sequence admits a subsequence which converges to some . This verifies and .
4.1.2 Algorithmic computability of entropy
Let a computable function. A real number is said to be computable with rate when there exists an algorithm which, given an integer as input, outputs in at most steps a rational number such that .
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 .
Theorem 2 ([Pavlov Schraudner]).
Let be a block-gluing bidimensional subshift of finite type. Then is computable with rate .
Let be a -block gluing bidimensional subshift of finite type on alphabet . For all , the number is equal to the number of patterns which appear in a locally-admissible rectangular pattern whose restrictions on the two extremal vertical (resp. horizontal) sides are equal.
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 and the total domination subshift satisfy the block-gluing property, with (just fill every cell with grey). In this section, we prove that and also satisfy this property.
In the following, for all , we denote by the column of .
The minimal domination subshift is block gluing and .
Idea of the proof: In order to simplify the proof 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 , 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.