Some Algorithmic Results for Eternal Vertex Cover Problem in Graphs

Abstract

Eternal vertex cover problem is a variant of the vertex cover problem. It is a two player (attacker and defender) game in which given a graph $$G=(V,E)$$ , the defender needs to allocate guards at some vertices so that the allocated vertices form a vertex cover. Attacker can attack one edge at a time and the defender needs to move the guards along the edges such that at least one guard moves through the attacked edge and the new configuration still remains a vertex cover. The attacker wins if no such move exists for the defender. The defender wins if there exists a strategy to defend the graph against infinite sequence of attacks. The minimum number of guards with which the defender can form a winning strategy is called the eternal vertex cover number of G, and is denoted by evc(G). Given a graph G, the problem of finding the eternal vertex cover number is NP-hard for general graphs, and remains NP-hard even for bipartite graphs. We have given a polynomial time algorithm to find the Eternal vertex cover number in chain graphs and cographs. We have also given a linear-time algorithm to find the eternal vertex cover number for split graphs, an important subclass of chordal graphs.