Diameter, short paths and superconnectivity in digraphs

Abstract

A connected digraph is said to be superconnected if it is maximally connected and every minimum disconnecting set F consists of the vertices adjacent to or from a given vertex not belonging to F. Let δ be the minimum degree of the digraph and π be a positive integer such that π ⩽ ⌊ δ / 2 ⌋ when δ ⩾ 7 , or π ⩽ ⌊ ( δ - 2 ) / 2 ⌋ for δ ⩾ 5 . We prove that G is maximally connected or has a good superconnectivity if the diameter D ⩽ 2 ℓ π - 2 and ℓ 0 ⩾ 2 , where ℓ π is a generalization of the semigirth ℓ 0 introduced by Fàbrega and Fiol (J. Graph Theory 13(6) (1989) 657). We also show that G is maximally connected if π ⩽ ⌊ ( δ - 1 ) / 2 ⌋ and 3 ⩽ δ ⩽ 6 . In the edge case, it is enough that D ⩽ 2 ℓ π - 1 . Finally, the obtained results are applied to the iterated line digraphs.