Results on the Small Quasi-Kernel Conjecture

J. Ai, S. Gerke, G. Gutin, Anders Yeo, Yacong Zhou
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引用次数: 2

Abstract

A {\em quasi-kernel} of a digraph $D$ is an independent set $Q\subseteq V(D)$ such that for every vertex $v\in V(D)\backslash Q$, there exists a directed path with one or two arcs from $v$ to a vertex $u\in Q$. In 1974, Chv\'{a}tal and Lov\'{a}sz proved that every digraph has a quasi-kernel. In 1976, Erd\H{o}s and S\'zekely conjectured that every sink-free digraph $D=(V(D),A(D))$ has a quasi-kernel of size at most $|V(D)|/2$. In this paper, we give a new method to show that the conjecture holds for a generalization of anti-claw-free digraphs. For any sink-free one-way split digraph $D$ of order $n$, when $n\geq 3$, we show a stronger result that $D$ has a quasi-kernel of size at most $\frac{n+3}{2} - \sqrt{n}$, and the bound is sharp.
关于小拟核猜想的结果
有向图{\em}$D$的核是一个独立的集合$Q\subseteq V(D)$,使得对于每个顶点$v\in V(D)\backslash Q$,存在一条从$v$到顶点$u\in Q$的有向路径,有一条或两条弧。1974年,Chvátal和Lovász证明了每个有向图都有一个拟核。1976年,Erd \H{o} s和Sźekely推测每个无汇有向图$D=(V(D),A(D))$都有一个大小不超过$|V(D)|/2$的拟核。本文给出了一种新的方法来证明反无爪有向图的一种推广的猜想成立。对于任意阶为$n$的无汇单向分裂有向图$D$,当$n\geq 3$时,我们给出了一个更强的结果,即$D$有一个大小不超过$\frac{n+3}{2} - \sqrt{n}$的拟核,并且界是锐利的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
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