2015 Volume E98.A Issue 6 Pages 1153-1160
Suppose one of the edges is attacked in a graph G, where some number of guards are placed on some of its vertices. If a guard is placed on one of the end-vertices of the attacked edge, she can defend such an attack to protect G by passing over the edge. For each of such attacks, every guard is allowed either to move to a neighboring vertex, or to stay at where she is. The eternal vertex cover number τ∞(G) is the minimum number of guards sufficient to protect G from any length of any sequence of edge attacks. This paper derives the eternal vertex cover number τ∞(G) of such graphs constructed by replacing each edge of a tree by an arbitrary elementary bipartite graph (or by an arbitrary clique), in terms of easily computable graph invariants only, thereby showing that τ∞(G) can be computed in polynomial time for such graphs G.