2 research outputs found
On the existence and number of -kings in -quasi-transitive digraphs
Let be a digraph and an integer. We say that
is -quasi-transitive if for every directed path in
, then or . Clearly, a
2-quasi-transitive digraph is a quasi-transitive digraph in the usual sense.
Bang-Jensen and Gutin proved that a quasi-transitive digraph has a 3-king
if and only if has a unique initial strong component and, if has a
3-king and the unique initial strong component of has at least three
vertices, then has at least three 3-kings. In this paper we prove the
following generalization: A -quasi-transitive digraph has a -king
if and only if has a unique initial strong component, and if has a
-king then, either all the vertices of the unique initial strong
components are -kings or the number of -kings in is at least
.Comment: 17 page