Superconnected digraphs and graphs with small conditional diameters

The conditional diameter Dν of a digraph G measures how far apart a pair of vertex sets V1 and V2 can be in such a way that the minimum out-degree and the minimum in-degree of the subdigraphs induced by V1 and V2, respectively, are at least ν. Thus, D0 is the standard diameter and D0 ≥ D1 ≥ . . . ≥ Dδ, where δ is the minimum degree. We prove that if Dν ≤ 2` − 3, where ` is a parameter related to the shortest paths, then G is maximally connected, superconnected or has a good superconnectivity, depending only on whether ν is equal to dδ/2e, d(δ− 1)/2e, d(δ− 1)/3e, respectively. In the edge case, it is enough that Dν ≤ 2` − 2. The results for graphs are obtained as a corollary of those for digraphs, because in the undirected case, ` = b(g − 1)/2c, g being the girth.