On some classes of quasi-triangulated graphs

IF 1 3区数学 Q3 MATHEMATICS, APPLIED

Discrete Applied Mathematics Pub Date : 2024-10-15 DOI:10.1016/j.dam.2024.09.029

Elaine M. Eschen , Chính T. Hoàng , R. Sritharan

{"title":"On some classes of quasi-triangulated graphs","authors":"Elaine M. Eschen , Chính T. Hoàng , R. Sritharan","doi":"10.1016/j.dam.2024.09.029","DOIUrl":null,"url":null,"abstract":"<div><div>A graph is quasi-triangulated if each of its induced subgraphs has a vertex that is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set) in the induced subgraph. The problem of characterizing quasi-triangulated graphs by minimal forbidden induced subgraphs is open. We show that a graph is quasi-triangulated if it does not contain as induced subgraph any of the graphs <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>, <span><math><mrow><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>3</mn></mrow></msub></mrow></math></span>, <em>fence</em>, or their complements. The fence is obtained from a <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> by substituting two adjacent vertices for each end-vertex of the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span> and two non-adjacent vertices for each interior vertex of the <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>. We also provide an efficient algorithm to recognize the subclass <span><math><msub><mrow><mi>Q</mi><mi>T</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for fixed <span><math><mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></math></span>, of quasi-triangulated graphs studied in the literature.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"359 ","pages":"Pages 419-424"},"PeriodicalIF":1.0000,"publicationDate":"2024-10-15","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/S0166218X24004190","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

A graph is quasi-triangulated if each of its induced subgraphs has a vertex that is either simplicial (its neighbours form a clique) or co-simplicial (its non-neighbours form an independent set) in the induced subgraph. The problem of characterizing quasi-triangulated graphs by minimal forbidden induced subgraphs is open. We show that a graph is quasi-triangulated if it does not contain as induced subgraph any of the graphs

C_{5}

P_{5}

P_{3} + P_{3}

, fence, or their complements. The fence is obtained from a

P_{4}

by substituting two adjacent vertices for each end-vertex of the

P_{4}

and two non-adjacent vertices for each interior vertex of the

P_{4}

. We also provide an efficient algorithm to recognize the subclass

{Q T}_{k}

, for fixed

k \geq 0

, of quasi-triangulated graphs studied in the literature.

查看原文本刊更多论文

关于准三角形图的一些类别

如果一个图的每个诱导子图都有一个顶点，而这个顶点在诱导子图中要么是简单的（它的邻居组成了一个小群），要么是共简单的（它的非邻居组成了一个独立集），那么这个图就是准三角形图。通过最小禁止诱导子图表征准三角形图的问题尚未解决。我们证明，如果一个图的诱导子图不包含 C5、P5、P3+P3、fence 或它们的补图，那么这个图就是准三角形图。用两个相邻顶点代替 P4 的每个末端顶点，用两个非相邻顶点代替 P4 的每个内部顶点，就可以从 P4 得到栅栏图。我们还提供了一种高效算法，用于识别文献中研究的准三角形图的子类 QTk（固定 k≥0）。

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.