A Linear Time Algorithm for Computing the Eternal Vertex Cover Number of Cactus Graphs

TL;DR

This paper introduces a linear time recursive algorithm to compute the eternal vertex cover number specifically for cactus graphs, leveraging a unique substructure property, and extends the approach to related graph classes.

Contribution

It presents the first linear time algorithm for eternal vertex cover number of cactus graphs based on a novel substructure property.

Findings

Linear time recursive algorithm for cactus graphs

Extension to graphs with blocks as edges, cycles, or biconnected chordal graphs

Efficient computation based on substructure property

Abstract

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.