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# A Result on the Small Quasi-Kernel Conjecture

Dec 2022

Any directed graph $D=(V(D),A(D))$ in this work is assumed to be finite andwithout self-loops. A source in a directed graph is a vertex having at leastone 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 avertex in $Q$. It is an open problem whether every source-free directed graphhas a quasi-kernel of size at most $|V(D)|/2$, a problem known as the smallquasi-kernel conjecture (SQKC). The aim of this paper is to prove the SQKCunder the assumption of a structural property of directed graphs. This relatesthe SQKC to the existence of a vertex $u\in V(D)$ and a bound on the number ofnew sources emerging when $u$ and its out-neighborhood are removed from $D$.The results in this work are of technical nature and therefore additionallyverified by means of the Coq proof-assistant.

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