On the existence and number of $(k+1)$-kings in $k$-quasi-transitive digraphs

TL;DR

This paper generalizes previous results on kings in quasi-transitive digraphs, establishing conditions for the existence and quantity of $(k+1)$-kings in $k$-quasi-transitive digraphs.

Contribution

It extends known theorems from quasi-transitive to $k$-quasi-transitive digraphs, characterizing when $(k+1)$-kings exist and their minimum number.

Findings

A $k$-quasi-transitive digraph has a $(k+1)$-king iff it has a unique initial strong component.

If a $(k+1)$-king exists, either all vertices in the initial component are $(k+1)$-kings or there are at least $(k+2)$ of them.

The results generalize previous findings for the case $k=2$ to arbitrary $k \

Abstract

Let $D=(V(D), A(D))$ be a digraph and $k \ge 2$ an integer. We say that $D$ is $k$-quasi-transitive if for every directed path $(v_0, v_1,..., v_k)$ in $D$, then $(v_0, v_k) \in A(D)$ or $(v_k, v_0) \in A(D)$. 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 $D$ has a 3-king if and only if $D$ has a unique initial strong component and, if $D$ has a 3-king and the unique initial strong component of $D$ has at least three vertices, then $D$ has at least three 3-kings. In this paper we prove the following generalization: A $k$-quasi-transitive digraph $D$ has a $(k+1)$-king if and only if $D$ has a unique initial strong component, and if $D$ has a $(k+1)$-king then, either all the vertices of the unique initial strong components are $(k+1)$-kings or the number of $(k+1)$-kings in $D$ is at least $(k+2)$.