On a Characterization of Spartan Graphs

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.