On a Characterization of Spartan Graphs

TL;DR

This work analyzes Spartan graphs, i.e., graphs with $evc(G)=mvc(G)$, in the eternal vertex cover game. It extends the known bipartite characterisation to König graphs and introduces a unifying, matching-based framework built on weakly/strongly good vertex covers and a mutual-reachability criterion encoded by the auxiliary graph $\mathfrak{h}_G(S,T)$. The authors prove a general, necessary-and-sufficient condition for $G$ to be Spartan via a non-empty family $\mathcal{F}$ of minimum vertex covers connected by vertex-disjoint guard-move paths, and they derive new lower bounds and a matching-based characterization that illuminate the structure of optimal defense strategies. The results significantly advance understanding of when minimal guard resources suffice and provide a practical criterion for verifying Spartan-ness across broad graph classes, including both König and general graphs.

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 König 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.

Figures (6)

• Figure 1: Examples of bad scenarios.
• Figure 2: Demonstrating reachability between vertex covers $S = \{a,c,e,g,h,i,j\}$ and $T = \{b,d,f,g,h,i,j\}$ with the vertex $\{k\}$ in the dead zone.
• Figure 3: The green wavy edges form a matching in the given graph $G$
• Figure 4: The purple wavy edges form a perfect matching in the given graph $G$
• Figure 5: The green wavy edges in the top figure do not form a maximum matching of the given graph $G$ although the matching is maximal, i.e., no other edge of the graph can be added to it to form a bigger matching. The red wavy edges in the bottom figure form a maximum matching of the given graph $G$.

Theorems & Definitions (26)

• Theorem 1
• Lemma 2
• Lemma 3
• Lemma 4
• Lemma 5
• Corollary 7
• Definition 8
• Definition 9
• Definition 10
• Definition 11