{"title":"弦图的相交和一些相关的分割问题","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":"{\"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}","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
摘要
一个图的和弦度是其交集为该图的和弦图的最小数目。可以利用扬纳卡基斯（Yannakakis）1982 年的一个结果来推断，对于每个固定的 k≥3，判断一个图的和弦度是否最多为 k 是 NP-complete。我们考虑的问题是判定一个图的和弦度是否为 2，或者等价于判定一个给定的图是否是两个和弦图的交集。我们证明，当其中一个弦图是分裂图，而另一个满足特定条件时，该问题等同于分割问题。利用这一点，我们推导出了各种问题的复杂性结果，包括判定一个图是否是 k 个分裂图的交集，对于 k=2 的问题，该问题在 P 级，而对于 k≥3 的问题，该问题在 NP 级。
Intersection of chordal graphs and some related partition problems
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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