On the Small Quasi-kernel conjecture

TL;DR

The paper surveys the theory of quasi-kernels in directed graphs, with a focus on the Small Quasi-kernel conjecture which asserts that any source-free digraph contains a quasi-kernel of size at most half the vertices. It analyzes the Chvátal-Lovász algorithm, its limitations, and constructions showing it need not produce small Qks, while also presenting reformulations and equivalences with the Kostochka-Luo-Shan conjecture. Key results include that KLS and the small Qk conjecture are equivalent, planarity-driven consequences via the four-color theorem, and a range of class-specific proofs; recent work extends to new bounds, kernel-like notions, and infinite-graph analogs. The survey also highlights open questions, structural conditions that guarantee small Qks, and the broader impact on understanding kernel-related questions in digraphs.

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.

Figures (1)

• Figure 1: A sketch of $G$; the set $A$ is a Qk of $G$, the set $B$ is out-neighborhood of $A$, and $C$ is the remaining set of vertices that are in the second out-neighborhood of $A$. We assume that $C$ has a kernel $K$. The sets $A$ and $B$ are partitioned into smaller subsets, as described in the proof of Theorem \ref{['th:good']}.

Theorems & Definitions (28)

• Theorem 1: Chvátal and Lovász, 1974, CL74
• Conjecture 2: Small Quasi-kernel conjecture Fete, 1976
• proof : Proof of Theorem \ref{['th:CL']}
• Theorem 3: Jacob and Meyniel, 1996
• Theorem 4: Gutin, Koh, Tay, and Yeo, 2004, gutin
• Theorem 5: Coitoru, CC15
• Theorem 6: Langlois, Meunier, Rizzi and Vialette LMRV21, 2021
• Example 7
• Lemma 8
• proof