On Vertices of outdegree n in minimally n‐connected digraphs

Abstract

AbstractLet |D| and |D|+n denote the number of vertices of D and the number of vertices of outdegree n in the digraph D, respectively. It is proved that every minimally n‐connected, finite digraph D has |D|+n ≥ n + 1 and that for n ≥ 2, there is a cn > 0 such that $|D|^+_n > C_n \sqrt{|D|}$ for all minimally n‐connected, finite digraphs D. Furthermore, case n = 2 of the following conjecture is settled which says that every minimally n‐connected, finite digraph has a vertex of indegree and outdegree equal to n. © 2002 John Wiley & Sons, Inc. J Graph Theory 39: 129–144, 2002