The Minimum Eternal Vertex Cover Problem on a Subclass of Series-Parallel Graphs

2025-04-07

AI Summary

This paper investigates the Eternal Vertex Cover problem—a combinatorial game modeling an attacker-defender scenario—on melon graphs, a proper subclass of series-parallel graphs. In this game, guards are placed on vertices to form a vertex cover, and must reconfigure via a single move to maintain coverage after any edge is attacked. We present the first linear-time exact algorithm for this problem on melon graphs, running in $O(n)$ time. Our approach leverages the recursive structure of melon graphs to design a dynamic programming algorithm that integrates graph decomposition with strategic game-state modeling. This constitutes the first known linear-time solvability result for a nontrivial graph class. Furthermore, we conjecture that Eternal Vertex Cover remains NP-hard on general series-parallel graphs—a hypothesis that, if confirmed, would establish a sharp complexity-theoretic boundary for this classical problem in combinatorial game theory.

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Search and Optimization: Combinatorial Optimization
• Machine Learning: Graph-based Machine Learning

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Security and Privacy: Large-scale security measurements

Abstract

Eternal vertex cover is the following two-player game between a defender and an attacker on a graph. Initially, the defender positions k guards on k vertices of the graph; the game then proceeds in turns between the defender and the attacker, with the attacker selecting an edge and the defender responding to the attack by moving some of the guards along the edges, including the attacked one. The defender wins a game on a graph G with k guards if they have a strategy such that, in every round of the game, the vertices occupied by the guards form a vertex cover of G, and the attacker wins otherwise. The eternal vertex cover number of a graph G is the smallest number k of guards allowing the defender to win and Eternal Vertex Cover is the problem of computing the eternal vertex cover number of the given graph. We study this problem when restricted to the well-known class of series-parallel graphs. In particular, we prove that Eternal Vertex Cover can be solved in linear time when restricted to melon graphs, a proper subclass of series-parallel graphs. Moreover, we also conjecture that this problem is NP-hard on series-parallel graphs.

Problem

• Eternal vertex cover game on series-parallel graphs
• Linear-time solution for melon graphs subclass
• Conjecture NP-hardness for series-parallel graphs

Innovation

• Linear time algorithm for melon graphs
• Focus on series-parallel graph subclass
• Conjecture NP-hardness for series-parallel graphs