ON THE CONNECTIVITY AND THE CONDITIONAL DIAMETER OF GRAPHS AND DIGRAPHS

Recently, it was proved that if the diameter D of a graph G is small e nough 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 less than or equal to 2l -1, the n G has maximum connectivity (kappa = delta), and if D less than or eq ual to 2l, then it attains maximum edge-connectivity (lambda = delta), 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 par ameter 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 ma ximum connectivity or edge-connectivity are derived. It is also shown that these conditions can be slightly relaxed when the digraphs are bi partite. The case of (undirected) graphs is managed as a corollary of the above results for digraphs. In particular, since I greater than or equal to 1, some known results of Plesnik and Znam are either reobtai ned or improved. For instance, it is shown that any graph whose line g raph has diameter D = 2 (respectively, D less than or equal to 3) has maximum connectivity (respectively, edge-connectivity.) Moreover, for graphs with even girth and minimum degree large enough, we obtain a lo wer bound on their connectivities. (C) 1996 John Wiley & Sons, Inc.