On graphs whose eternal vertex cover number and vertex cover number coincide

Abstract

The eternal vertex cover problem is a variant of the classical vertex cover problem defined in terms of an infinite attacker-defender game played on a graph. In each round of the game, the defender reconfigures guards from one vertex cover to another in response to a move by the attacker. The minimum number of guards required in any winning strategy of the defender when this game is played on a graph G is the eternal vertex cover number of G, denoted by evc(G). It is known that given a graph G and an integer k, checking whether evc(G) ≤ k is NP-hard. Further, it is known that for any graph G, mvc(G) ≤ evc(G) ≤ 2 mvc(G), where mvc(G) is the vertex cover number of G. Though a characterization is known for graphs for which evc(G) = 2 mvc(G), a characterization of graphs for which evc(G) = mvc(G) remained as an open problem, since 2009. We achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For biconnected chordal graphs, our characterization leads to a polynomial time algorithm for precisely determining evc(G) and an algorithm for determining a safe strategy for guard movement in each round of the game using only evc(G) guards. Though the eternal vertex cover problem is only known to be in PSPACE in general, it follows from our new characterization that the problem is in NP for locally connected graphs, a graph class which includes all biconnected internally triangulated planar graphs. We also provide reductions establishing NP-completeness of the problem for biconnected internally triangulated planar graphs. As far as we know, this is the first NP-completeness result known for the problem for any graph class.

A new lower bound for eternal vertex cover number

“…This improved lower bound has many algorithmic consequences, including a quadratic time algorithm for computing the eternal vertex cover number of chordal graphs and a PTAS for computing the eternal vertex cover number of internally triangulated planar graphs. These results generalize the results presented in [2].…”

Eternal vertex cover number of maximal outerplanar graphs

“…The techniques presented in the paper make crucial use of the planarity of the underlying graph to yield a divide and conquer algorithm for computing the evc number of a maximal outerplanar graph. Attempts to generalize the techniques to maximal planar graphs may not be successful due to the known NP hardness result on the evc computation of biconnected internally triangulated planar graphs [2]. However, the complexity status of the problem of computing the evc number of outerplanar graphs is open and may be attempted using the techniques developed in this work.…”

“…Consequent to the hardness results, work on solving the problem on various graph classes have been attempted in the literature. It is known that the problem is NP-complete even for biconnected internally triangulated planar graphs [2]. Polynomial time algorithms for computing evc(G) were known exactly only for very elementary graph classes such as an O(n) algorithm for trees [3], a polynomial time algorithm for a tree-like graph class [4] and a linear time algorithm for cactus graphs [5].…”

Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

“…Klostermeyer and Mynhardt (2009) gave a characterization of the graphs for which the upper bound is achieved. A characterization for graphs for which lower bound is achieved remains open, but several special cases have been addressed in the literature (see, for instance Babu et al, 2021). Also, Klostermeyer and Mynhardt (2011) study graphs with equal eternal vertex cover and eternal domination numbers, which is a closely related dynamic variant of the dominating set problem.…”

“…The comptuational question of ETERNAL VERTEX COVER is also well studied on special classes of graphs. For instance, it is known to be NP-complete when restricted to locally connected graphs, a graph class which includes all biconnected internally triangulated planar graphs (Babu et al, 2021). It can also be solved in linear time on the class of cactus graphs , quadratic time on chordal graphs and in polynomial time on "generalized" trees (Araki et al, 2015).…”