  The eternal vertex cover problem is a dynamic variant of the classical vertex
cover problem. It is NP-hard to compute the eternal vertex cover number of
graphs and known algorithmic results for the problem are very few. This paper
presents a linear time recursive algorithm for computing the eternal vertex
cover number of cactus graphs. Unlike other graph classes for which polynomial
time algorithms for eternal vertex cover number are based on efficient
computability of a known lower bound directly derived from minimum vertex
cover, we show that it is a certain substructure property that helps the
efficient computation of eternal vertex cover number of cactus graphs. An
extension of the result to graphs in which each block is an edge, a cycle or a
biconnected chordal graph is also presented.
