Enumeration of maximum matchings of graphs

TL;DR

This paper presents a new formula and algorithm for counting maximum matchings in general graphs using Gallai-Edmonds structure, extending previous results and enabling enumeration of maximum matchings efficiently.

Contribution

It introduces a novel formula based on Gallai-Edmonds decomposition for counting maximum matchings and provides an algorithm for enumeration, extending prior work on trees.

Findings

Derived a computing formula for maximum matchings using Gallai-Edmonds theorem.

Developed an algorithm to enumerate maximum matchings based on the formula.

Extended previous results to general graphs and applied to opt trees.

Abstract

Counting maximum matchings in a graph is of great interest in statistical mechanics, solid-state chemistry, theoretical computer science, mathematics, among other disciplines. However, it is a challengeable problem to explicitly determine the number of maximum matchings of general graphs. In this paper, using Gallai-Edmonds structure theorem, we derive a computing formula for the number of maximum matching in a graph. According to the formula, we obtain an algorithm to enumerate maximum matchings of a graph. In particular, The formula implies that computing the number of maximum matchings of a graph is converted to compute the number of perfect matchings of some induced subgraphs of the graph. As an application, we calculate the number of maximum matchings of opt trees. The result extends a conclusion obtained by Heuberger and Wagner[C. Heuberger, S. Wagner, The number of maximum matchings in a tree, Discrete Math. 311 (2011) 2512--2542].