In this paper we investigate the following generalization of transitivity: A digraph D is (m,n)-transitive whenever there is a path of length m from x to y there is a subset of n+1 vertices of these m+1 vertices which contain a path of length n from x to y.
Here we study various properties of (m,n)-transitive digraphs. In particular, (m,1)-transitive tournaments are characterized. Their similarities to transitive tournaments are analyzed and discussed.
Various other results pertaining to (m,1)-transitive digraphs are given.
