Let be a graph, which is assuming throughout to be simple and finite. A set is called a dominating set if for every there exists an such that . Consider the following graph security model. A set of guards begins by occupying a dominating set in a graph , and must respond to an infinite sequence of attacks. By this, we mean that after a vertex is chosen by an attacker, one guard which is adjacent to must move from its current position to ; necessarily, the resulting set of positions must still be a dominating set of . If guards can move in this way to respond to any infinite sequence of attacks, then we say can be eternally -guarded. The minimum for which can be eternally -guarded is called the eternal domination number of , which is denoted .
The study of was introduced in [burger2004infinite], where (among other topics) its relation to other graph parameters was studied. Recall that an independent set of is a set of pairwise non-adjacent vertices, and , called the independence number of , is the maximum cardinality of an independent set in . A clique of is a set of pairwise adjacent vertices. A clique cover of is a set for which each is a clique and . The cardinality of a minimum clique cover of is called the clique cover number of and is denoted . It was shown in [burger2004infinite] that, for any graph , . Many open questions remain regarding eternal domination with respect to these bounding parameters, and in particular whether or not particular constructions force to be equal to one of them (the survey [survey] provides a nice overview of what is and is not known about eternal domination and its variants).
We make use of two graph binary operations on graphs in this paper. The join of two graphs and , denoted , is the graph obtained by adding all possible edges between and . The Cartesian product of two graphs and , denoted , has and an edge between and if and only if either and or and . The product ( denotes the complete graph on vertices) is called the prism of ; one can informally think of the prism of as the graph obtained by taking two copies of and adding a matching between corresponding vertices. In [KM15], the following conjecture is put forward:
[KM15] If is a graph such that , then .
The purpose of this paper is to present a construction of an infinite family of counterexamples to this conjecture.
2 A Mycielskian construction
Throughout the paper, we use to denote the set . The Mycielskian of a graph , with , denoted , is defined as follows:
This construction was introduced in [mycielski1955coloriage], the purpose of which was to demonstrate the existence of triangle-free graphs with arbitrarily large chromatic number (recall that the chromatic number of a graph , denoted , is the minimum number of colours needed to colour so that no adjacent vertices receive the same colour). In particular, Mycielski proved the following:
[mycielski1955coloriage] For any graph , .
The following simple lemma follows directly from the construction of :
For any connected graph on at least two vertices, .
Recall that a graph is called vertex-critical if for every .
If is a vertex-critical graph, then is vertex-critical.
Let and let be a minimum proper -colouring of . We proceed by cases on the vertex to be deleted from , showing in each case that . If , then we assign . Suppose . Since is vertex critical, we may assume that is a unique colour in . In this case can be extended to by letting for and . Finally, suppose that , and that is unique in . In this case, can be extended by assigning for all , and may be given any colour used in colouring which is distinct from . ∎
We now define a particular family of graphs obtained by the Mycielskian operation. Let be an integer and let be a family of graphs defined recursively by and for each integer , . By Lemmas 2 and 3, and . By applying simple induction, we also have the following:
For each , is vertex-critical.
3 Main result
The construction of our family of counterexamples to Conjecture 1 requires the following three lemmas.
[klostermeyer2007eternal] For any graph , .
[KM05] Let be a graph such that , , . If is an integer such that , then , and .
Let be a graph and let denote the number of cliques of cardinality in a clique cover of . If is the maximum value of taken over all clique covers of size , then .
Without loss of generality, suppose the vertex set of is . Let be a clique cover of and let . For any , if is the subgraph induced by the vertices of the set then is a clique cover of . For any , if then is a clique of and covers the vertices and . Without loss of generality, suppose each of the cliques covers a unique vertex and each of the cliques have at least two vertices. The set covers , hence . Observe that a clique of is contained in either or or the clique is of the form . The cliques contained in (respectively ) with the vertices (respectively ) in cover (respectively ). Therefore, we then have the inverse inequality. ∎
We now use the construction of to obtain our infinite family of counterexamples, which may have arbitrarily large independence number, eternal domination number, and clique cover number.
For any integer , there exists a graph such that and .
Let be an integer greater than or equal to , let , and let . Since and , we have that and . Since is vertex-critical (Lemma 5), for any given vertex there exists a proper -coloring of such that receives a unique color. As a consequence, for any given vertex , there exists a clique covering of by a minimum number of cliques where the clique which contains is of cardinality . By Lemma 6, we have that . Let ; by Lemma 7, and . Observe that for any given vertex , there exists a minimum clique covering of such that the clique which contains is of cardinality , since each vertex of can be added to a distinct clique of not containing . Fix to be any vertex in , and note that has a maximum independent set for which . Finally, let be the graph obtained from by adding a single vertex adjacent only to . Then, . By taking the minimum clique cover of for which is the only singleton, we may extend it to a clique cover of containing no singletons by replacing with ; by Lemma 8, this implies that . On the other hand, denoting , guards can protect the subgraph induced by the vertices in the set , guards can protect the subgraph induced by the vertices in the set and one guard can protect the vertices and . Thus . ∎
Partial financial support for this work was received from the Fonds de recherche du Québec– Nature et technologies and from the Natural Sciences and Engineering Research Council of Canada.