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Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

Neeldhara Misra, Saraswati Nanoti

TL;DR

This paper studies Eternal Vertex Cover, a dynamic variant of vertex cover where an attacker targets edges and a defender reconfigures guards to defend, formalized by the parameter $evc(G)$. It proves that the problem is NP-hard and does not admit a polynomial kernel on bipartite graphs of diameter six via a reduction from Red Blue Dominating Set, while also providing a polynomial-time algorithm for cobipartite graphs by characterizing when $evc(G)$ equals $mvc(G)$ or $evc(G)=p+q-1$ in terms of the bipartition structure. The results close the complexity gap for bipartite graphs and deliver a complete structural understanding for cobipartite graphs, with implications for Eternal Connected Vertex Cover and kernelization barriers. The work also discusses potential improvements in fixed-parameter tractability and approximation approaches for these graph classes.

Abstract

Eternal Vertex Cover problem is a dynamic variant of the vertex cover problem. We have a two player game in which guards are placed on some vertices of a graph. In every move, one player (the attacker) attacks an edge. In response to the attack, the second player (defender) moves the guards along the edges of the graph in such a manner that at least one guard moves along the attacked edge. If such a movement is not possible, then the attacker wins. If the defender can defend the graph against an infinite sequence of attacks, then the defender wins. The minimum number of guards with which the defender has a winning strategy is called the Eternal Vertex Cover Number of the graph G. On general graphs, the computational problem of determining the minimum eternal vertex cover number is NP-hard and admits a 2-approximation algorithm and an exponential kernel. The complexity of the problem on bipartite graphs is open, as is the question of whether the problem admits a polynomial kernel. We settle both these questions by showing that Eternal Vertex Cover is NP-hard and does not admit a polynomial compression even on bipartite graphs of diameter six. We also show that the problem admits a polynomial time algorithm on the class of cobipartite graphs.

Eternal Vertex Cover on Bipartite and Co-Bipartite Graphs

TL;DR

This paper studies Eternal Vertex Cover, a dynamic variant of vertex cover where an attacker targets edges and a defender reconfigures guards to defend, formalized by the parameter . It proves that the problem is NP-hard and does not admit a polynomial kernel on bipartite graphs of diameter six via a reduction from Red Blue Dominating Set, while also providing a polynomial-time algorithm for cobipartite graphs by characterizing when equals or in terms of the bipartition structure. The results close the complexity gap for bipartite graphs and deliver a complete structural understanding for cobipartite graphs, with implications for Eternal Connected Vertex Cover and kernelization barriers. The work also discusses potential improvements in fixed-parameter tractability and approximation approaches for these graph classes.

Abstract

Eternal Vertex Cover problem is a dynamic variant of the vertex cover problem. We have a two player game in which guards are placed on some vertices of a graph. In every move, one player (the attacker) attacks an edge. In response to the attack, the second player (defender) moves the guards along the edges of the graph in such a manner that at least one guard moves along the attacked edge. If such a movement is not possible, then the attacker wins. If the defender can defend the graph against an infinite sequence of attacks, then the defender wins. The minimum number of guards with which the defender has a winning strategy is called the Eternal Vertex Cover Number of the graph G. On general graphs, the computational problem of determining the minimum eternal vertex cover number is NP-hard and admits a 2-approximation algorithm and an exponential kernel. The complexity of the problem on bipartite graphs is open, as is the question of whether the problem admits a polynomial kernel. We settle both these questions by showing that Eternal Vertex Cover is NP-hard and does not admit a polynomial compression even on bipartite graphs of diameter six. We also show that the problem admits a polynomial time algorithm on the class of cobipartite graphs.
Paper Structure (6 sections, 8 theorems, 1 equation, 4 figures)

This paper contains 6 sections, 8 theorems, 1 equation, 4 figures.

Table of Contents

  1. Introduction
  2. Our Contributions
  3. Preliminaries and Notations
  4. Hardness on Bipartite Graphs
  5. A Polynomial-time Algorithm for Co-bipartite Graphs
  6. Concluding Remarks

Key Result

Proposition 4

Let $P$ and $Q$ be parameterized problems such that there is a polynomial parameter transformation from $P$ to $Q$. If $Q$ has a polynomial compression, then $P$ also has a polynomial compression.

Figures (4)

  • Figure 1: An attack that is defended by moving two guards.
  • Figure 7: An overview of the proof of Theorem 2.
  • Figure 8: An overview of the cases in the proof of Theorem 2 (continued from \ref{['fig:overview']}).
  • Figure 13: This figure demonstrates Case $3c III$ from \ref{['big']}.

Theorems & Definitions (12)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • Proposition 5: Corollary 19.6 kernelbook
  • Lemma 6
  • Claim 10
  • Lemma 11
  • Lemma 12
  • Lemma 13
  • ...and 2 more