Quasi-kernels in split graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics Pub Date : 2024-10-30 DOI:10.1016/j.dam.2024.10.009

Hélène Langlois , Frédéric Meunier , Romeo Rizzi , Stéphane Vialette , Yacong Zhou

{"title":"Quasi-kernels in split graphs","authors":"Hélène Langlois , Frédéric Meunier , Romeo Rizzi , Stéphane Vialette , Yacong Zhou","doi":"10.1016/j.dam.2024.10.009","DOIUrl":null,"url":null,"abstract":"<div><div>In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The “small quasi-kernel conjecture”, proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a <span><math><mrow><mn>1</mn><mo>/</mo><mn>2</mn></mrow></math></span> ratio, but even with larger ratio, this property is known to hold only for few classes of graphs.</div><div>The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph <span><math><mi>D</mi></math></span> has a quasi-kernel of size at most <span><math><mrow><mfrac><mrow><mn>2</mn></mrow><mrow><mn>3</mn></mrow></mfrac><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>D</mi><mo>)</mo></mrow><mo>|</mo></mrow></mrow></math></span>, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"361 ","pages":"Pages 236-243"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Applied Mathematics","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0166218X24004335","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}

引用次数: 0

Abstract

In a digraph, a quasi-kernel is a subset of vertices that is independent and such that the shortest path from every vertex to this subset is of length at most two. The “small quasi-kernel conjecture”, proposed by Erdős and Székely in 1976, postulates that every sink-free digraph has a quasi-kernel whose size is within a fraction of the total number of vertices. The conjecture is even more precise with a

1 / 2

ratio, but even with larger ratio, this property is known to hold only for few classes of graphs.

The focus here is on small quasi-kernels in split graphs. This family of graphs has played a special role in the study of the conjecture since it was used to disprove a strengthening that postulated the existence of two disjoint quasi-kernels. The paper proves that every sink-free split digraph

D

has a quasi-kernel of size at most

\frac{2}{3} | V (D) |

, and even of size at most two when the graph is an orientation of a complete split graph. It is also shown that computing a quasi-kernel of minimal size in a split digraph is W[2]-hard.

查看原文本刊更多论文

分裂图中的准核

在一个数图中，准核是一个独立的顶点子集，使得从每个顶点到这个子集的最短路径的长度最多为 2。Erdős 和 Székely 于 1976 年提出了 "小准核猜想"，假设每个无汇数图都有一个准核，其大小在顶点总数的几分之一以内。这一猜想在比率为 1/2 时更为精确，但即使比率更大，这一性质也只在少数几类图中成立。这个图族在该猜想的研究中发挥了特殊作用，因为它被用来推翻假设存在两个不相交准核的强化。本文证明了每个无汇分裂数图 D 最多有一个大小为 23|V(D)| 的准核，当该图是一个完整分裂图的定向时，甚至最多有一个大小为两个的准核。同时还证明，计算一个分裂数图中最小尺寸的准核是 W[2]-hard 的。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

求助全文

约1分钟内获得全文求助全文

来源期刊

Discrete Applied Mathematics 数学-应用数学

CiteScore

2.30

自引率

9.10%

发文量

422

审稿时长

4.5 months

期刊介绍： The aim of Discrete Applied Mathematics is to bring together research papers in different areas of algorithmic and applicable discrete mathematics as well as applications of combinatorial mathematics to informatics and various areas of science and technology. Contributions presented to the journal can be research papers, short notes, surveys, and possibly research problems. The "Communications" section will be devoted to the fastest possible publication of recent research results that are checked and recommended for publication by a member of the Editorial Board. The journal will also publish a limited number of book announcements as well as proceedings of conferences. These proceedings will be fully refereed and adhere to the normal standards of the journal. Potential authors are advised to view the journal and the open calls-for-papers of special issues before submitting their manuscripts. Only high-quality, original work that is within the scope of the journal or the targeted special issue will be considered.