Virgelot Virgile (Universite de Montreal)
Eternal domination on cartesian product of graphs with $K_2$

The eternal domination problem is a graph protection model where mobile guards are placed on some vertices of a graph and these guards must move to adjacent vertices in order to respond to a sequence of attacks on the vertices of that graph. In this talk, we consider the "one guard moves" model in which exactly one guard is allowed to move to respond to a single attack at discrete time. The eternal domination number of a graph, which is the minimum number of guards required to protect the graph against an infinite number of attacks, is closely related to the independence number and the clique cover number of the graph. Klostermeyer and Mynhardt conjectured that if the eternal domination number of a graph is equal to the clique cover number of that graph, then the eternal domination number of the cartesian product of the graph with K_2 is equal to the clique cover number of that cartesian product. We will refute the conjecture by providing an infinite class of counterexample.