On the Small Quasi-kernel conjecture

TL;DR

This paper surveys results on quasi-kernels in directed graphs, focusing on the small quasi-kernel conjecture, which posits the existence of small quasi-kernels in graphs without zero in-degree vertices, and presents new proofs and results.

Contribution

It provides a comprehensive survey of quasi-kernel research, explores the small quasi-kernel conjecture, and introduces new proofs and findings related to the conjecture.

Findings

Every directed graph has a quasi-kernel that can be found in linear time.

The small quasi-kernel conjecture holds for certain classes of graphs.

New proofs and partial results supporting the conjecture are presented.

Abstract

An independent vertex subset $S$ of the directed graph $G$ is a kernel if the set of out-neighbors of $S$ is $V(G)\setminus S$. An independent vertex subset $Q$ of $G$ is a quasi-kernel if the union of the first and second out-neighbors contains $V(G)\setminus S$ as a subset. Deciding whether a directed graph has a kernel is an NP-hard problem. In stark contrast, each directed graph has quasi-kernel(s) and one can be found in linear time. In this article, we will survey the results on quasi-kernel and their connection with kernels. We will focus on the small quasi-kernel conjecture which states that if the graph has no vertex of zero in-degree, then there exists a quasi-kernel of size not larger than half of the order of the graph. The paper also contains new proofs and some new results as well.