Making Bipartite Graphs DM-irreducible

The Dulmage–Mendelsohn decomposition (or the DM-decomposition) gives a unique partition of the vertex set of a bipartite graph reflecting the structure of all the maximum matchings therein. A bipartite graph is said to be DM-irreducible if its DM-decomposition consists of a single component. For connected bipartite graphs, this is equivalent to the condition that every edge is contained in some maximum matching. In this paper, we focus on the problem of making a given bipartite graph DM-irreducible by adding edges. When the input bipartite graph is balanced (i.e., both sides have the same number of vertices) and has a perfect matching, this problem is equivalent to making a directed graph strongly connected by adding edges, for which the minimum number of additional edges was characterized by Eswaran and Tarjan (1976). We give a general solution to this problem, which is divided mainly into two parts. When the input graph is unbalanced, the problem is solved via matroid intersection. This result can be extended to the minimum cost version in which each potential edge has individual cost to be added. For balanced input graphs, we devise a combinatorial algorithm that finds a minimum number of additional edges to attain the DM-irreducibility, while the minimum cost version of this problem is NP-hard. Both results also lead to similar minmax characterizations of the minimum number, which generalize the result of Eswaran and Tarjan. We also show that our problem can be formulated as a special case of the bisupermodular covering problem introduced by Frank and Jordán (1995). This provides an alternative proof to our min-max characterization. Eötvös Loránd University, 1117 Budapest, Hungary. Email: [email protected] University of Tokyo, Tokyo 113-8656, Japan. Email: [email protected] Toyota Motor Corporation, Toyota 471-8571, Japan. Email: jun kato [email protected] Osaka University, Osaka 565-0871, Japan. Email: yutaro [email protected]