Diameter, short paths and superconnectivity in digraphs

A connected digraph of minimum degree δ is said to be maximally connected if the cardinality of every cutset is al least δ, and it is called superconnected if moreover every minimum disconnecting set F consists of the vertices adjacent to or from a given vertex not belonging to F . Let π be an integer, 0 ≤ π ≤ δ − 2, δ being the minimum degree of the digraph. In this work, we prove that if D ≤ 2` − 2 and ` ≥ 2, where ` is a parameter related to the shortest paths, then G is maximally connected or has a good superconnectivity depending only on whether π ≤ bδ/2c and δ ≥ 3, or π ≤ b(δ − 2)/2c and δ ≥ 5, respectively. In the edge case, it is enough that D ≤ 2` − 1. Finally, the obtained results are applied to the iterated line digraphs.