Mengerian graphs: characterization and recognition (2208.06517v1)

Abstract: A temporal graph ${\cal G}$ is a graph that changes with time. More specifically, it is a pair $(G, \lambda)$ where $G$ is a graph and $\lambda$ is a function on the edges of $G$ that describes when each edge $e\in E(G)$ is active. Given vertices $s,t\in V(G)$, a temporal $s,t$-path is a path in $G$ that traverses edges in non-decreasing time; and if $s,t$ are non-adjacent, then a temporal $s,t$-cut is a subset $S\subseteq V(G)\setminus{s,t}$ whose removal destroys all temporal $s,t$-paths. It is known that Menger's Theorem does not hold on this context, i.e., that the maximum number of internally vertex disjoint temporal $s,t$-paths is not necessarily equal to the minimum size of a temporal $s,t$-cut. In a seminal paper, Kempe, Kleinberg and Kumar (STOC'2000) defined a graph $G$ to be Mengerian if equality holds on $(G,\lambda)$ for every function $\lambda$. They then proved that, if each edge is allowed to be active only once in $(G,\lambda)$, then $G$ is Mengerian if and only if $G$ has no gem as topological minor. In this paper, we generalize their result by allowing edges to be active more than once, giving a characterization also in terms of forbidden structures. We additionally provide a polynomial time recognition algorithm.