A Result on the Small Quasi-Kernel Conjecture

TL;DR

This paper proves the small quasi-kernel conjecture for directed graphs under certain structural assumptions, linking the conjecture to vertex removal properties and verifying results with formal proof assistance.

Contribution

It establishes the small quasi-kernel conjecture under a specific structural property, connecting it to vertex removal and source emergence in directed graphs.

Findings

Proves the small quasi-kernel conjecture under a structural assumption.

Links the conjecture to the existence of a special vertex and source emergence bounds.

Results are verified using the Coq proof-assistant.

Abstract

Any directed graph $D=(V(D),A(D))$ in this work is assumed to be finite and without self-loops. A source in a directed graph is a vertex having at least one ingoing arc. A quasi-kernel $Q\subseteq V(D)$ is an independent set in $D$ such that every vertex in $V(D)$ can be reached in at most two steps from a vertex in $Q$. It is an open problem whether every source-free directed graph has a quasi-kernel of size at most $|V(D)|/2$, a problem known as the small quasi-kernel conjecture (SQKC). The aim of this paper is to prove the SQKC under the assumption of a structural property of directed graphs. This relates the SQKC to the existence of a vertex $u\in V(D)$ and a bound on the number of new sources emerging when $u$ and its out-neighborhood are removed from $D$. The results in this work are of technical nature and therefore additionally verified by means of the Coq proof-assistant.