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On Graphs with Minimal Eternal Vertex Cover Number

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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Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Indian Institute of Technology PalakkadPalakkadIndia
  2. 2.Indian Institute of ScienceBangaloreIndia
  3. 3.Indian Statistical InstituteChennaiIndia
  4. 4.National Institute of Technology CalicutCalicutIndia

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