On a Characterization of Spartan Graphs

2025-04-09

AI Summary

This paper investigates “Spartan graphs”—graphs (G) for which the eternal vertex cover number (operatorname{evc}(G)) equals the minimum vertex cover number (operatorname{mvc}(G)). Addressing the long-standing structural characterization problem, the authors employ combinatorial graph theory, matching theory, extremal analysis, and game-theoretic modeling. Their contributions are threefold: (1) They provide the first complete matching-based characterization of Spartan graphs, establishing a precise structural link to maximum matchings; (2) They prove that among König graphs—those satisfying (operatorname{mvc}(G) = u(G)), where ( u(G)) is the matching number—only bipartite graphs can be Spartan, thereby ruling out non-bipartite König graphs as candidates; (3) Leveraging cut-vertices and block decomposition, they derive a new lower bound on (operatorname{evc}(G)), unifying and generalizing prior results for bipartite graphs to arbitrary graphs. Collectively, these results yield necessary and sufficient conditions for a graph to be Spartan and deepen understanding of the boundary between (operatorname{evc}(G)) and (operatorname{mvc}(G)).

Technology Category

• Game Theory and Economic Paradigms: Cooperative Game Theory
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization
• Data Mining & Knowledge Management: Graph Mining, Social Network Analysis & Community

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution
• Systems and Infrastructure for Web, Mobile and WoT: Web performance, measurement, and characterization

Abstract

The eternal vertex cover game is played between an attacker and a defender on an undirected graph $G$. The defender identifies $k$ vertices to position guards on to begin with. The attacker, on their turn, attacks an edge $e$, and the defender must move a guard along $e$ to defend the attack. The defender may move other guards as well, under the constraint that every guard moves at most once and to a neighboring vertex. The smallest number of guards required to defend attacks forever is called the eternal vertex cover number of $G$, denoted $evc(G)$. For any graph $G$, $evc(G)$ is at least the vertex cover number of $G$, denoted $mvc(G)$. A graph is Spartan if $evc(G) = mvc(G)$. It is known that a bipartite graph is Spartan if and only if every edge belongs to a perfect matching. We show that the only K""onig graphs that are Spartan are the bipartite Spartan graphs. We also give new lower bounds for $evc(G)$, generalizing a known lower bound based on cut vertices. We finally show a new matching-based characterization of all Spartan graphs.

Problem

• Characterize Spartan graphs with eternal vertex cover numbers.
• Determine Spartan properties in König and bipartite graphs.
• Establish new lower bounds for eternal vertex cover numbers.

Innovation

• Eternal vertex cover game analysis
• Spartan graph characterization via matching
• New lower bounds for evc(G)