Recently, it was proved that if the diameter D of a graph G is small enough in comparison with its girth, then G is maximally connected and that a similar result also holds for digraphs. More precisely, if the diameter D of a digraph G satisfies D ⩽ 2I – 1, then G has maximum connectivity (κ = δ), and if D ⩽ 2I, then it attains maximum edge-connectivity (λ = δ), where I is a parameter which can be thought of as a generalization of the girth of a graph. In this paper, we study some similar conditions for a digraph to attain high connectivities, which are given in terms of what we call the conditional diameter or P-diameter of G. This parameter measures how far apart can be a pair of subdigraphs satisfying a given property P, and, hence, it generalizes the standard concept of diameter. As a corollary, some new sufficient conditions to attain maximum connectivity or edge-connectivity are derived. It is also shown that these conditions can be slightly relaxed when the digraphs are bipartite. The case of (undirected) graphs is managed as a corollary of the above results for digraphs. In particular, since I ⩾ 1, some known results of Plesnik and Znám are either reobtained or improved. For instance, it is shown that any graph whose line graph has diameter D = 2 (respectively, D ⩽ 3) has maximum connectivity (respectively, edge-connectivity). Moreover, for graphs with even girth and minimum degree large enough, we obtain a lower bound on their connectivities.
