Kings in quasi-transitive digraphs

A k -king in a digraph D is a vertex which can reach every other vertex by a directed path of length at most k . This definition generalizes the definition of a king in a tournament. We consider quasi-transitive digraphs - a generalization of tournaments recently investigated by the authors (Bang-Jensen and Huang, 1995). We prove that a quasi-transitive digraph has a 3-king if and only if it has an out-branching. We give several results on 3-kings in quasi-transitive digraphs which are analogous to well-known results about kings in tournaments.