On existence of noncritical vertices in digraphs

TL;DR

This paper investigates conditions under which strongly connected digraphs contain noncritical vertices, establishing degree sum bounds that guarantee the existence of one or two such vertices, with tightness demonstrated through examples.

Contribution

It proves new degree sum conditions that ensure the existence of noncritical vertices in strongly connected digraphs, extending understanding of their structural properties.

Findings

If the sum of degrees of any two adjacent vertices is at least n+1, a noncritical vertex exists.

If the sum of degrees of any two adjacent vertices is at least n+2, two noncritical vertices exist.

The bounds provided are shown to be tight through examples.

Abstract

Let $D$ be a strongly connected digraphs on $n\ge 4$ vertices. A vertex $v$ of $D$ is noncritical, if the digraph $D-v$ is strongly connected. We prove, that if sum of the degrees of any two adjacent vertices of $D$ is at least $n+1$, then there exists a noncritical vertex in $D$, and if sum of the degrees of any two adjacent vertices of $D$ is at least $n+2$, then there exist two noncritical vertices in $D$. A series of examples confirm that these bounds are tight.