On the Small Quasi-kernel Conjecture—A Survey

Erdős, Péter and Győri, Ervin and Mezei, Tamás Róbert and Salia, Nika and Tyomkyn, M. (2025) On the Small Quasi-kernel Conjecture—A Survey. In: Lecture Notes in Computer Science. Lecture Notes in Computer Science (15795). Springer Science and Business Media Deutschland GmbH, pp. 98-110. ISBN 9783031932236

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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)\\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)\\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. (A preliminary version of this survey was published as [8]). © The Author(s), under exclusive license to Springrer Nature Switzerland AG 2025.

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