# On Graphs with Minimal Eternal Vertex Cover Number

• Jasine Babu
• L. Sunil Chandran
• Mathew Francis
• Veena Prabhakaran
• J. Nandini Warrier
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11394)

## Abstract

The eternal vertex cover problem is a variant of the classical vertex cover problem where a set of guards on the vertices have to be dynamically reconfigured from one vertex cover to another in every round of an attacker-defender game. The minimum number of guards required to protect a graph from an infinite sequence of attacks is the eternal vertex cover number (evc) of the graph. It is known that, given a graph G and an integer k, checking whether $${\text {evc}}(G) \le k$$ is NP-Hard. However, for any graph G, $${\text {mvc}}(G) \le {\text {evc}}(G) \le 2 {\text {mvc}}(G)$$, where $${\text {mvc}}(G)$$ is the minimum vertex cover number of G. Precise value of eternal vertex cover number is known only for certain very basic graph classes like trees, cycles and grids. Though a characterization is known for graphs for which $${\text {evc}}(G) = 2{\text {mvc}}(G)$$, a characterization of graphs for which $${\text {evc}}(G) = {\text {mvc}}(G)$$ remained open. Here, we achieve such a characterization for a class of graphs that includes chordal graphs and internally triangulated planar graphs. For some graph classes including biconnected chordal graphs, our characterization leads to a polynomial time algorithm to precisely determine $${\text {evc}}(G)$$ and to determine a safe strategy of guard movement in each round of the game with $${\text {evc}}(G)$$ guards.

## Keywords

Eternal vertex cover Chordal graphs Connected vertex cover

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© Springer Nature Switzerland AG 2019

## Authors and Affiliations

• Jasine Babu
• 1
Email author
• L. Sunil Chandran
• 2
• Mathew Francis
• 3
• Veena Prabhakaran
• 1
• 1
• J. Nandini Warrier
• 4
2. 2.Indian Institute of ScienceBangaloreIndia
3. 3.Indian Statistical InstituteChennaiIndia
4. 4.National Institute of Technology CalicutCalicutIndia