Excluding a Planar Matching Minor in Bipartite Graphs

TL;DR

This paper extends the theory of graph minors to bipartite graphs with perfect matchings, establishing a link between planarity, matching minors, and algorithmic tractability for problems like counting perfect matchings.

Contribution

It introduces a matching minor version of the Erdos-Posa property, characterizes bipartite graphs with bounded perfect matching width, and provides polynomial algorithms for related problems.

Findings

Matching minors relate to planarity in bipartite graphs.

Graphs excluding a planar matching minor have bounded perfect matching width.

Counting perfect matchings is polynomial-time solvable for graphs excluding a fixed planar matching minor.

Abstract

Matching minors are a specialisation of minors fit for the study of graph with perfect matchings. The notion of matching minors has been used to give a structural description of bipartite graphs on which the number of perfect matchings can becomputed 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 introducea version of Erdos-Posa property for matching minors and find a direct link between this property and planarity. From this, it follows that a class of bipartite graphs withperfect matchings has bounded perfect matching width if and only if it excludes aplanar 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 graphGcontains afixed planar graphHas 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.