Bisimplicial separators

IF 0.9 3区 数学 Q2 MATHEMATICS
Martin Milanič, Irena Penev, Nevena Pivač, Kristina Vušković
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For an integer <span></span><math>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>0</mn>\n </mrow></math>, we say that a minimal separator is <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math>-<i>simplicial</i> if it can be covered by <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math> cliques and denote by <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>k</mi>\n </msub>\n </mrow></math> the class of all graphs in which each minimal separator is <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math>-simplicial. We show that for each <span></span><math>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>0</mn>\n </mrow></math>, the class <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>k</mi>\n </msub>\n </mrow></math> is closed under induced minors, and we use this to show that the \n<span>Maximum Weight Stable Set</span> problem can be solved in polynomial time for <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>k</mi>\n </msub>\n </mrow></math>. We also give a complete list of minimal forbidden induced minors for <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mn>2</mn>\n </msub>\n </mrow></math>. Next, we show that, for <span></span><math>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>1</mn>\n </mrow></math>, every nonnull graph in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>k</mi>\n </msub>\n </mrow></math> has a <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math>-simplicial vertex, that is, a vertex whose neighborhood is a union of <span></span><math>\n \n <mrow>\n <mi>k</mi>\n </mrow></math> cliques; we deduce that the \n<span>Maximum Weight Clique</span> problem can be solved in polynomial time for graphs in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mn>2</mn>\n </msub>\n </mrow></math>. Further, we show that, for <span></span><math>\n \n <mrow>\n <mi>k</mi>\n \n <mo>≥</mo>\n \n <mn>3</mn>\n </mrow></math>, it is NP-hard to recognize graphs in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mi>k</mi>\n </msub>\n </mrow></math>; the time complexity of recognizing graphs in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mn>2</mn>\n </msub>\n </mrow></math> is unknown. We also show that the \n<span>Maximum Clique</span> problem is NP-hard for graphs in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mn>3</mn>\n </msub>\n </mrow></math>. Finally, we prove a decomposition theorem for diamond-free graphs in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mn>2</mn>\n </msub>\n </mrow></math> (where the <i>diamond</i> is the graph obtained from <span></span><math>\n \n <mrow>\n <msub>\n <mi>K</mi>\n \n <mn>4</mn>\n </msub>\n </mrow></math> by deleting one edge), and we use this theorem to obtain polynomial-time algorithms for the \n<span>Vertex Coloring</span> and recognition problems for diamond-free graphs in <span></span><math>\n \n <mrow>\n <msub>\n <mi>G</mi>\n \n <mn>2</mn>\n </msub>\n </mrow></math>, and improved running times for the \n<span>Maximum Weight Clique</span> and \n<span>Maximum Weight Stable Set</span> problems for this class of graphs.</p>","PeriodicalId":16014,"journal":{"name":"Journal of Graph Theory","volume":"106 4","pages":"816-842"},"PeriodicalIF":0.9000,"publicationDate":"2024-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.23098","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Graph Theory","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jgt.23098","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0

Abstract

A minimal separator of a graph G is a set S V ( G ) such that there exist vertices a , b V ( G ) S with the property that S separates a from b in G , but no proper subset of S does. For an integer k 0 , we say that a minimal separator is k -simplicial if it can be covered by k cliques and denote by G k the class of all graphs in which each minimal separator is k -simplicial. We show that for each k 0 , the class G k is closed under induced minors, and we use this to show that the  Maximum Weight Stable Set problem can be solved in polynomial time for G k . We also give a complete list of minimal forbidden induced minors for G 2 . Next, we show that, for k 1 , every nonnull graph in G k has a k -simplicial vertex, that is, a vertex whose neighborhood is a union of k cliques; we deduce that the  Maximum Weight Clique problem can be solved in polynomial time for graphs in G 2 . Further, we show that, for k 3 , it is NP-hard to recognize graphs in G k ; the time complexity of recognizing graphs in G 2 is unknown. We also show that the  Maximum Clique problem is NP-hard for graphs in G 3 . Finally, we prove a decomposition theorem for diamond-free graphs in G 2 (where the diamond is the graph obtained from K 4 by deleting one edge), and we use this theorem to obtain polynomial-time algorithms for the  Vertex Coloring and recognition problems for diamond-free graphs in G 2 , and improved running times for the  Maximum Weight Clique and  Maximum Weight Stable Set problems for this class of graphs.

Abstract Image

二等分隔符
一个图的最小分隔符是这样一个集合,即存在这样的顶点,它们具有从 、 中分隔开来的属性,但没有适当的子集这样做。对于整数 ,如果最小分隔符可以被小块覆盖,我们就说它是-简单的,并用所有每个最小分隔符都是-简单的图的类来表示。我们证明,对于每个 ,该类在诱导最小分隔符下是封闭的,并以此证明最大权重稳定集问题可以在多项式时间内求解.我们还给出了......的最小禁止诱导最小数的完整列表。接下来,我们证明,对于......中的每个非空图,都有一个简单顶点,也就是说,有一个顶点的邻域是小群的联合;我们推导出,对于......中的图,最大权重小群问题可以在多项式时间内求解。此外,我们还证明了,对于......,识别......中的图是 NP 难的;识别......中的图的时间复杂度尚不清楚。我们还证明,对于......中的图,最大克立(Maximum Clique)问题是 NP-hard。最后,我们证明了 in 中无菱形图的分解定理(其中菱形图是通过删除一条边得到的图),并利用该定理得到了针对 , 中无菱形图的顶点着色问题和识别问题的多项式时间算法,以及针对该类图的最大权重簇问题和最大权重稳定集问题的改进运行时间。
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来源期刊
Journal of Graph Theory
Journal of Graph Theory 数学-数学
CiteScore
1.60
自引率
22.20%
发文量
130
审稿时长
6-12 weeks
期刊介绍: The Journal of Graph Theory is devoted to a variety of topics in graph theory, such as structural results about graphs, graph algorithms with theoretical emphasis, and discrete optimization on graphs. The scope of the journal also includes related areas in combinatorics and the interaction of graph theory with other mathematical sciences. A subscription to the Journal of Graph Theory includes a subscription to the Journal of Combinatorial Designs .
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