Mengerian Temporal Graphs Revisited

Abstract

A temporal graph $\mathcal{G}$ 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 vertex temporal $s,t$-cut is a subset $S\subseteq V(G)$ 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 vertex 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 also provide a polynomial time recognition algorithm.

Supported by CNPq grants 303803/2020-7 and 437841/2018-9, FUNCAP/CNPq grant PNE-0112-00061.01.00/16, and CAPES (PhD student funding).