Matching Cuts in Graphs of High Girth and H-Free Graphs

IF 0.7 4区计算机科学 Q4 COMPUTER SCIENCE, SOFTWARE ENGINEERING

Algorithmica Pub Date : 2025-05-08 DOI:10.1007/s00453-025-01318-8

Carl Feghali, Felicia Lucke, Daniël Paulusma, Bernard Ries

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引用次数: 0

Abstract

The (Perfect) Matching Cut problem is to decide if a connected graph has a (perfect) matching that is also an edge cut. The Disconnected Perfect Matching problem is to decide if a connected graph has a perfect matching that contains a matching cut. Both Matching Cut and Disconnected Perfect Matching are NP-complete for planar graphs of girth 5, whereas Perfect Matching Cut is known to be NP-complete even for subcubic bipartite graphs of arbitrarily large fixed girth. We prove that Matching Cut and Disconnected Perfect Matching are also NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Our result for Matching Cut resolves a 20-year old open problem. We also show that the more general problem d-Cut, for every fixed \(d\ge 1\), is NP-complete for bipartite graphs of arbitrarily large fixed girth and bounded maximum degree. Furthermore, we show that Matching Cut, Perfect Matching Cut and Disconnected Perfect Matching are NP-complete for H-free graphs whenever H contains a connected component with two vertices of degree at least 3. Afterwards, we update the state-of-the-art summaries for H-free graphs and compare them with each other, and with a known and full classification of the Maximum Matching Cut problem, which is to determine a largest matching cut of a graph G. Finally, by combining existing results, we obtain a complete complexity classification of Perfect Matching Cut for \(\mathcal{H}\)-subgraph-free graphs where \(\mathcal{H}\) is any finite set of graphs.

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高周长图与无h图的匹配切

（完美）匹配切问题是决定一个连通图是否有一个（完美）匹配也是一个切边。断开的完美匹配问题是决定一个连通图是否有一个包含匹配切割的完美匹配。对于周长为5的平面图，匹配切割和断开完美匹配都是np完全的，而对于任意大的固定周长的亚三次二部图，完美匹配切割也是np完全的。我们证明了对于任意大固定周长和有界最大度的二部图，匹配切割和断开完美匹配也是np完全的。我们匹配切割的结果解决了一个20年的开放性问题。我们还证明了更一般的问题d-Cut，对于每一个固定\(d\ge 1\)，对于任意大的固定周长和有界的最大度的二部图是np完全的。进一步，我们证明了匹配切割、完美匹配切割和断开完美匹配对于H-free图是np完全的，当H包含两个顶点至少为3度的连通分量时。随后，我们更新了无h图的最新总结，并将它们相互比较，并与已知的最大匹配切问题的完整分类进行了比较，该问题是确定图g的最大匹配切。最后，通过结合现有结果，我们获得了\(\mathcal{H}\) -子图无图的完美匹配切的完整复杂性分类，其中\(\mathcal{H}\)是任意有限图集。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Algorithmica 工程技术-计算机：软件工程

CiteScore

2.80

自引率

9.10%

发文量

158

审稿时长

12 months

期刊介绍： Algorithmica is an international journal which publishes theoretical papers on algorithms that address problems arising in practical areas, and experimental papers of general appeal for practical importance or techniques. The development of algorithms is an integral part of computer science. The increasing complexity and scope of computer applications makes the design of efficient algorithms essential. Algorithmica covers algorithms in applied areas such as: VLSI, distributed computing, parallel processing, automated design, robotics, graphics, data base design, software tools, as well as algorithms in fundamental areas such as sorting, searching, data structures, computational geometry, and linear programming. In addition, the journal features two special sections: Application Experience, presenting findings obtained from applications of theoretical results to practical situations, and Problems, offering short papers presenting problems on selected topics of computer science.