Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 4702387, 7 pages

http://dx.doi.org/10.1155/2016/4702387

## Maximum Matchings of a Digraph Based on the Largest Geometric Multiplicity

College of Information Engineering, Taiyuan University of Technology, Taiyuan, Shanxi 030024, China

Received 6 December 2015; Accepted 10 April 2016

Academic Editor: Zhike Peng

Copyright © 2016 Yunyun Yang and Gang Xie. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Matching theory is one of the most forefront issues of graph theory. Based on the largest geometric multiplicity, we develop an efficient approach to identify maximum matchings in a digraph. For a given digraph, it has been proved that the number of maximum matched nodes has close relationship with the largest geometric multiplicity of the transpose of the adjacency matrix. Moreover, through fundamental column transformations, we can obtain the matched nodes and related matching edges. In particular, when a digraph contains a cycle factor, the largest geometric multiplicity is equal to one. In this case, the maximum matching is a perfect matching and each node in the digraph is a matched node. The method is validated by an example.

#### 1. Introduction

Matching theory is one of the fundamental branches of graph theory. Historically, matching not only can be used for us to understand the structure of a graph which plays a crucial role but also has been related to a wide range of important problems in theoretical aspects, such as combinatorial optimization, crystal physics, and theoretical computer science research. In addition, matching theory is also closely linked with practical problems in work arrangements, resource allocation, information transmission, network flow, transportation and postservice, and so forth. To solve these problems we need to seek the maximum matching which is one of the core issues in matching theory [1–3]. Not only is it interesting by itself, but it can also be used to solve an army of other problems in combinatorial optimization [4–6]. Therefore, the work of the maximum matching theory has profound theoretical significance and wide application background.

The maximum matching in an undirected graph is a maximum set of edges without common nodes. The book by Burkard et al. [7] reviewed thoroughly the bipartite matching problem. The popular classic method using the Hopcroft-Karp algorithm needs the determination of the bipartite equivalent graph. For arbitrary graphs, it is complicated to find a maximum matching and Edmonds and Karp [8] presented a polynomial algorithm. It was a major breakthrough and innovation to find a maximum matching. Additionally, on this basis, a multitude of algorithms are given in the literatures [9–13]. Subsequently, the work [14] gave a few maximum matching problems such as the maximum-cardinality matching problem and the minimum cost perfect matching problem. Later, Dobson et al. [15] developed a new kind of maximum matching graphs. They pointed out maximum matchings of special graphs such as trees, cycles, or complete graphs. Reference [16] presented minimization of the Laplacian spectral radius of trees with given matching number. Moreover, Duarte et al. [17] established a method to maintain maximum matching under addition and deletion of edges in a graph. Even et al. [18] then proposed a method of computing rough estimate maximum-cardinality matchings and estimation maximum weight matchings based on deterministic distributed algorithms. The work in [19] introduced an algorithm to calculate maximum matchings in a bipartite graph.

However, the abovementioned study was focused on undirected graphs. Relatively speaking, for the maximum matching of digraphs only a little research has been conducted. Additionally, the solution of the matching is mainly based on the bipartite graph method, while the solution of the bipartite graph itself remains a tough issue. So we are still in lack of mature theories and algorithms. Therefore, we put forward a new method to identify maximum matchings in a digraph. The main idea of the method stems from two conditions of controllability of complex networks [20, 21]. The process of the method is simple, yet effective. In a sense, the work of this paper has developed and enriched matching theory.

The rest of the paper is organized as follows. Section 2 introduces the basic notations and definitions used in this paper. In Section 3, the technique in the case of a digraph is presented. The first part of Section 3 describes the technical concept based on minimum input theorem [20] and the largest geometric multiplicity theorem [21], and the second part of it introduces the algorithm steps. Section 4 presents an example to validate the method and Section 5 gives the conclusion and future work.

#### 2. Notations and Preliminaries

For integrity, we give a few notations and definitions in this section.

Let be a digraph which consists of a nonempty finite set of elements called nodes and a finite set of ordered pairs of different nodes called edges. We write . The size of is the number of nodes in , denoted by . In Figure 1, for instance, , .