On the order and size of s-geodetic digraphs with given connectivity

A digraph G = (V, E) with diameter D is said to be s-geodetic, for 1 ≤ s ≤ D, if between any pair of (not necessarily different) vertices x, y ∈ V there is at most one x → y path of length ≤ s. Thus, any loopless digraph is at least 1-geodetic. A similar definition applies for a graph G, but in this case the concept is closely related to its girth g, for then G is s-geodetic with s = b(g − 1)/2c. The case s = D corresponds to the so-called (strongly) geodetic (di)graphs. Some recent results have shown that if the order n of a (di)graph is big enough, then its vertex connectivity attains its maximum value. In other words, the (di)graph is maximally connected. Moreover, a similar result involving the size m (number of edges) and edge-connectivity applies. In this work we mainly show that the same conclusions can be reached if the order or size of a s-geodetic (di)graph is small enough. As a corollary, we find some Chartrand-type conditions to assure maximum connectivities. For example, when s ≥ 2, a s-geodetic digraph is maximally connected if δ ≥ ⌈ s n2 − 1 ⌉ . Under similar hypotheses it is also shown that stronger measures of connectivity, such as the so-called super-connectivity, attain also their maximum possible values.