{"title":"Intersection of chordal graphs and some related partition problems","authors":"","doi":"10.1016/j.dam.2024.10.010","DOIUrl":null,"url":null,"abstract":"<div><div>The chordality of a graph is the minimum number of chordal graphs whose intersection is the graph. A result of Yannakakis’ from 1982 can be used to infer that for every fixed <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, deciding whether the chordality of a graph is at most <span><math><mi>k</mi></math></span> is NP-complete. We consider the problem of deciding whether the chordality of a graph is 2, or equivalently, deciding whether a given graph is the intersection of two chordal graphs. We prove that the problem is equivalent to a partition problem when one of the chordal graphs is a split graph and the other meets certain conditions. Using this we derive complexity results for a variety of problems, including deciding if a graph is the intersection of <span><math><mi>k</mi></math></span> split graphs, which is in P for <span><math><mrow><mi>k</mi><mo>=</mo><mn>2</mn></mrow></math></span> and NP-complete for <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":null,"pages":null},"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/S0166218X24004360","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
The chordality of a graph is the minimum number of chordal graphs whose intersection is the graph. A result of Yannakakis’ from 1982 can be used to infer that for every fixed , deciding whether the chordality of a graph is at most is NP-complete. We consider the problem of deciding whether the chordality of a graph is 2, or equivalently, deciding whether a given graph is the intersection of two chordal graphs. We prove that the problem is equivalent to a partition problem when one of the chordal graphs is a split graph and the other meets certain conditions. Using this we derive complexity results for a variety of problems, including deciding if a graph is the intersection of split graphs, which is in P for and NP-complete for .
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.
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