Mengerian graphs: Characterization and recognition

A temporal graph G is a pair (G,λ) where G is a graph and λ is a function on the edges of G describing when each edge is active. Temporal connectivity then concerns only paths that respect the flow of time. In this context, it is known that Menger's Theorem does not hold. In a seminal paper, Kempe, Kleinberg and Kumar (STOC'2000) defined a graph to be Mengerian if equality holds for every time-function. They then proved that, if each edge is allowed to be active only once in (G,λ), 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.