Polygon-circle and word-representable graphs

Q2 Mathematics

Electronic Notes in Discrete Mathematics Pub Date : 2019-03-01 DOI:10.1016/j.endm.2019.02.001

Jessica Enright, Sergey Kitaev

{"title":"Polygon-circle and word-representable graphs","authors":"Jessica Enright, Sergey Kitaev","doi":"10.1016/j.endm.2019.02.001","DOIUrl":null,"url":null,"abstract":"<div>We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.A graph G = (V, E) is word-representable if there exists a word w over the alpha-bet V such that letters x and y form a subword of the form xyxy ⋯ or yxyx ⋯ iff xy is an edge in E. Word-representable graphs generalise several well-known and well-studied classes of graphs [S. Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, Lecture Notes in Computer Science 10396 (2017) 36–67; S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015]. It is known that any word-representable graph is k-word-representable, that is, can be represented by a word having exactly k copies of each letter for some k dependent on the graph. Recognising whether a graph is word-representable is NP-complete ([S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015, Theorem 4.2.15]). A polygon-circle graph (also known as a spider graph) is the intersection graph of a set of polygons inscribed in a circle [M. Koebe, On a new class of intersection graphs, Ann. Discrete Math. (1992) 141–143]. That is, two vertices of a graph are adjacent if their respective polygons have a non-empty intersection, and the set of polygons that correspond to vertices in this way are said to represent the graph. Recognising whether an input graph is a polygon-circle graph is NP-complete [M. Pergel, Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, Graph-Theoretic Concepts in Computer Science: 33rd Int. Workshop, Lecture Notes in Computer Science, 4769 (2007) 238–247]. We show that neither of these two classes is included in the other one by showing that the word-representable Petersen graph and crown graphs are not polygon-circle, while the non-word-representable wheel graph W5 is polygon-circle. We also provide a more refined result showing that for any k ≥ 3, there are k-word-representable graphs which are neither (k −1)-word-representable nor polygon-circle.</div>","PeriodicalId":35408,"journal":{"name":"Electronic Notes in Discrete Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2019-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://sci-hub-pdf.com/10.1016/j.endm.2019.02.001","citationCount":"3","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Electronic Notes in Discrete Mathematics","FirstCategoryId":"1085","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S1571065319300010","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"Mathematics","Score":null,"Total":0}

引用次数: 3

Abstract

We describe work on the relationship between the independently-studied polygon-circle graphs and word-representable graphs.

A graph G = (V, E) is word-representable if there exists a word w over the alpha-bet V such that letters x and y form a subword of the form xyxy ⋯ or yxyx ⋯ iff xy is an edge in E. Word-representable graphs generalise several well-known and well-studied classes of graphs [S. Kitaev, A Comprehensive Introduction to the Theory of Word-Representable Graphs, Lecture Notes in Computer Science 10396 (2017) 36–67; S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015]. It is known that any word-representable graph is k-word-representable, that is, can be represented by a word having exactly k copies of each letter for some k dependent on the graph. Recognising whether a graph is word-representable is NP-complete ([S. Kitaev, V. Lozin, “Words and Graphs”, Springer, 2015, Theorem 4.2.15]). A polygon-circle graph (also known as a spider graph) is the intersection graph of a set of polygons inscribed in a circle [M. Koebe, On a new class of intersection graphs, Ann. Discrete Math. (1992) 141–143]. That is, two vertices of a graph are adjacent if their respective polygons have a non-empty intersection, and the set of polygons that correspond to vertices in this way are said to represent the graph. Recognising whether an input graph is a polygon-circle graph is NP-complete [M. Pergel, Recognition of polygon-circle graphs and graphs of interval filaments is NP-complete, Graph-Theoretic Concepts in Computer Science: 33rd Int. Workshop, Lecture Notes in Computer Science, 4769 (2007) 238–247]. We show that neither of these two classes is included in the other one by showing that the word-representable Petersen graph and crown graphs are not polygon-circle, while the non-word-representable wheel graph W₅ is polygon-circle. We also provide a more refined result showing that for any k ≥ 3, there are k-word-representable graphs which are neither (k −1)-word-representable nor polygon-circle.

查看原文本刊更多论文

多边形-圆和文字可表示的图形

我们描述了独立研究的多边形圆图和可词表示图之间的关系。图G = (V, E)是词可表示的，如果在字母V上存在一个词w，使得字母x和y形成xyxy⋯或yxyx⋯形式的子词，如果xy是E中的边，则图G = (V, E)是词可表示的。词可表示的图概括了几种众所周知且研究得很好的图类[S]。基塔耶夫，《可词表示图理论概论》，《计算机科学讲义》(2017)36-67;S. Kitaev, V. Lozin，《Words and Graphs》，Springer, 2015。我们知道，任何一个可词表示的图都是k个词可表示的，也就是说，对于依赖于图的某个k，每个字母都有k个副本的一个词可以表示。识别图是否可词表示是np完全的([S]。Kitaev, V. Lozin，“Words and Graphs”，Springer, 2015, Theorem 4.2.15)。多边形-圆图(又称蜘蛛图)是一组内嵌在圆内的多边形的交点图[M]。关于一类新的交图，安。离散数学。(1992) 141 - 143]。也就是说，如果一个图的两个顶点各自的多边形有一个非空的交点，那么它们就是相邻的，并且以这种方式对应于顶点的多边形集被称为表示该图。识别输入图是否为多边形-圆图是np完全的[M]。潘杰，多边形-圆图和区间细丝图的np完全识别，计算机科学图论概念，第33卷。《计算机科学》，vol . 11(2007): 238-247。我们通过证明可词表示的Petersen图和crown图不是多边形-圆，而不可词表示的wheel图W5是多边形-圆来证明这两类都不包括在另一类中。我们还提供了一个更精细的结果，表明对于任何k≥3，存在k字可表示的图，这些图既不是(k−1)字可表示的，也不是多边形-圆。

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

Electronic Notes in Discrete Mathematics Mathematics-Discrete Mathematics and Combinatorics

CiteScore

1.30

自引率

0.00%

发文量

期刊介绍： Electronic Notes in Discrete Mathematics is a venue for the rapid electronic publication of the proceedings of conferences, of lecture notes, monographs and other similar material for which quick publication is appropriate. Organizers of conferences whose proceedings appear in Electronic Notes in Discrete Mathematics, and authors of other material appearing as a volume in the series are allowed to make hard copies of the relevant volume for limited distribution. For example, conference proceedings may be distributed to participants at the meeting, and lecture notes can be distributed to those taking a course based on the material in the volume.