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

TL;DR

The paper investigates the Minimum Eternal Vertex Cover problem on series-parallel graphs, with a focus on melon graphs. It proves that $evc(G)$ can be computed in linear time for melon graphs by a detailed, constructive case analysis based on the parity of the constituent $s$–$t$ paths and explicit defense strategies. The authors classify melon graphs into odd, even, and mixed types and provide tight, configuration-based defenses showing $evc(G)=vc(G)$ for odd melons and $evc(G)=vc(G)+1$ for even and mixed melons, culminating in a linear-time algorithm. They also conjecture NP-hardness for the broader SP class, supported by structural contrasts (notably the unbounded alt($G$)) and explicit G_k constructions that separate $evc$ from $vc$ by a growing margin, guiding future inquiries on SP graphs and related outerplanar subclasses.

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.

Figures (5)

• Figure 1: For each graph in the figure, the black vertices show a configuration of a minimum eternal vertex cover class, the red edge is the attacked one, and the arrows highlight the movement of the guards. Figures a. and b.: odd melon graph, the strategy described in the proof of \ref{['lem:odd']} according to the two cases of the proof of \ref{['lem:elem']}. Figures c. and d.: even melon graph, the two cases in the proof of \ref{['lem:evenmelon']}.
• Figure 2: For each graph in the figure, the black vertices show a configuration of a minimum eternal vertex cover class. Figures a., b. and c.: mixed melon graph with at least two even paths and two odd paths. Figure a. highlights even internal and external paths, and odd $s$- and $t$-paths. In Figures b. and c. the red edge is the attacked one, and the arrows highlight the movement of the guards, two cases in the proof of \ref{['lem:mixedmelonboth']}.
• Figure 3: For each graph in the figure, the black vertices show a configuration of a minimum eternal vertex cover class, the red edge is the attacked one, and the arrows highlight the movement of the guards. Figures a. and b.: mixed melon graph with at least two even paths and only one odd path, the two cases in the proof of \ref{['lem:mixedmelonsingleo']}.
• Figure 4: For each graph in the figure, the black vertices show a configuration of a minimum eternal vertex cover class. Figures a. and b.: mixed melon graph with at least two odd paths and only one even path, configurations $U_s$ and $U_{\mathcal{S}_o}$, respectively, used in the proof of \ref{['lem:mixedmelonsinglee']}.
• Figure 5: The figure shows the series-parallel graph $G_3$ described in the proof of \ref{['lem:unbounded']}. The black vertices represent its unique minimum vertex cover $U$. The red vertices are an example of the position of guards to be added to $U$ in order to get an eternal vertex cover configuration $U$.

Theorems & Definitions (19)

• Theorem 1
• Conjecture
• Definition 2
• Definition 3
• Definition 4
• Definition 5
• Definition 6
• Lemma 7
• Lemma 8
• Theorem 9