{"title":"Dichotomies for Maximum Matching Cut: H-freeness, bounded diameter, bounded radius","authors":"","doi":"10.1016/j.tcs.2024.114795","DOIUrl":null,"url":null,"abstract":"<div><p>The <span>(Perfect) Matching Cut</span> problem is to decide if a graph <em>G</em> has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of <em>G</em>. Both <span>Matching Cut</span> and <span>Perfect Matching Cut</span> are known to be <span>NP</span>-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the <span>Maximum Matching Cut</span> problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of <span>Maximum Matching Cut</span> for graphs of bounded diameter, bounded radius and <em>H</em>-free graphs. A disconnected perfect matching of a graph <em>G</em> is a perfect matching that contains a matching cut of <em>G</em>. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem <span>Disconnected Perfect Matching</span> is polynomial-time solvable for <span><math><mo>(</mo><msub><mrow><mi>P</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>+</mo><mi>s</mi><msub><mrow><mi>P</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span>-free graphs for every <span><math><mi>s</mi><mo>≥</mo><mn>0</mn></math></span>, extending a known result for <span><math><msub><mrow><mi>P</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graphs (Bouquet and Picouleau, 2020).</p></div>","PeriodicalId":49438,"journal":{"name":"Theoretical Computer Science","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0304397524004122/pdfft?md5=080f5be8832cd8e314696c9f4da5ed04&pid=1-s2.0-S0304397524004122-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Theoretical Computer Science","FirstCategoryId":"94","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0304397524004122","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"COMPUTER SCIENCE, THEORY & METHODS","Score":null,"Total":0}
引用次数: 0
Abstract
The (Perfect) Matching Cut problem is to decide if a graph G has a (perfect) matching cut, i.e., a (perfect) matching that is also an edge cut of G. Both Matching Cut and Perfect Matching Cut are known to be NP-complete. A perfect matching cut is also a matching cut with maximum number of edges. To increase our understanding of the relationship between the two problems, we perform a complexity study for the Maximum Matching Cut problem, which is to determine a largest matching cut in a graph. Our results yield full dichotomies of Maximum Matching Cut for graphs of bounded diameter, bounded radius and H-free graphs. A disconnected perfect matching of a graph G is a perfect matching that contains a matching cut of G. We also show how our new techniques can be used for finding a disconnected perfect matching with a largest matching cut for special graph classes. In this way we can prove that the decision problem Disconnected Perfect Matching is polynomial-time solvable for -free graphs for every , extending a known result for -free graphs (Bouquet and Picouleau, 2020).
完美)匹配切分问题是判断图 G 是否有(完美)匹配切分,即(完美)匹配也是 G 的边切分。完美匹配切分也是具有最大边数的匹配切分。为了加深我们对这两个问题之间关系的理解,我们对最大匹配切分问题进行了复杂性研究,即确定图中最大的匹配切分。我们的研究结果得出了最大匹配切分对有界直径图、有界半径图和无 H 图的完全二分法。图 G 的断开完美匹配是包含 G 的匹配切点的完美匹配。我们还展示了如何利用我们的新技术为特殊图类找到具有最大匹配切点的断开完美匹配。通过这种方法,我们可以证明,对于每 s≥0 的 (P6+sP2)-free 图,决策问题断开完美匹配是多项式时间可解的,从而扩展了针对无 P5 图的已知结果(Bouquet 和 Picouleau,2020 年)。
期刊介绍:
Theoretical Computer Science is mathematical and abstract in spirit, but it derives its motivation from practical and everyday computation. Its aim is to understand the nature of computation and, as a consequence of this understanding, provide more efficient methodologies. All papers introducing or studying mathematical, logic and formal concepts and methods are welcome, provided that their motivation is clearly drawn from the field of computing.