Excluding a planar matching minor in bipartite graphs

IF 16.4 1区化学 Q1 CHEMISTRY, MULTIDISCIPLINARY

Accounts of Chemical Research Pub Date : 2023-10-04 DOI:10.1016/j.jctb.2023.09.003

Archontia C. Giannopoulou , Stephan Kreutzer , Sebastian Wiederrecht

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引用次数: 4

Abstract

The notion of matching minors is a specialisation of minors fit for the study of graphs with perfect matchings. Matching minors have been used to give a structural description of bipartite graphs on which the number of perfect matchings can be computed efficiently, based on a result of Little, by McCuaig et al. in 1999.

In this paper we generalise basic ideas from the graph minor series by Robertson and Seymour to the setting of bipartite graphs with perfect matchings. We introduce a version of Erdős-Pósa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs with perfect matchings has bounded perfect matching width if and only if it excludes a planar matching minor. We also present algorithms for bipartite graphs of bounded perfect matching width for a matching version of the disjoint paths problem, matching minor containment, and for counting the number of perfect matchings. From our structural results, we obtain that recognising whether a bipartite graph G contains a fixed planar graph H as a matching minor, and that counting the number of perfect matchings of a bipartite graph that excludes a fixed planar graph as a matching minor are both polynomial time solvable.

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二部图中排除平面匹配子

匹配未成年人的概念是适合研究具有完美匹配的图的未成年人的专业化。在McCuaig等人1999年的Little结果的基础上，匹配子被用于给出二分图的结构描述，在该结构描述上可以有效地计算完全匹配的数量。本文将Robertson和Seymour的图子级数的基本思想推广到具有完全匹配的二分图设置。我们介绍了一个用于匹配未成年人的Erdõs-Pósa属性的版本，并找到了该属性与平面性之间的直接联系。由此可知，一类具有完全匹配的二部图具有有界的完全匹配宽度，当且仅当它排除了一个平面匹配子。我们还提出了有界完全匹配宽度的二分图的算法，用于不相交路径问题的匹配版本、匹配小包容以及计算完全匹配的数量。从我们的结构结果中，我们得到了识别二分图G是否包含固定平面图H作为匹配子，以及计算排除固定平面图作为匹配子的二分图的完全匹配次数都是多项式时间可解的。

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

Accounts of Chemical Research 化学-化学综合

CiteScore

31.40

自引率

1.10%

发文量

312

审稿时长

2 months

期刊介绍： Accounts of Chemical Research presents short, concise and critical articles offering easy-to-read overviews of basic research and applications in all areas of chemistry and biochemistry. These short reviews focus on research from the author’s own laboratory and are designed to teach the reader about a research project. In addition, Accounts of Chemical Research publishes commentaries that give an informed opinion on a current research problem. Special Issues online are devoted to a single topic of unusual activity and significance. Accounts of Chemical Research replaces the traditional article abstract with an article "Conspectus." These entries synopsize the research affording the reader a closer look at the content and significance of an article. Through this provision of a more detailed description of the article contents, the Conspectus enhances the article's discoverability by search engines and the exposure for the research.