Connected matchings in chordal bipartite graphs

A connected matching in a graph is a collection of edges that are pairwise disjoint but joined by another edge of the graph. Motivated by applications to Hadwiger’s conjecture, Plummer, Stiebitz, and Toft [13] introduced connected matchings and proved that, given a positive integer k, determining whether a graph has a connected matching of size at least k is NPcomplete. Cameron [4] proved that this problem remains NP-complete on bipartite graphs, but can be solve in polynomial-time on chordal graphs. We present a polynomial-time algorithm that finds a maximum connected matching in a chordal bipartite graph. This includes a novel edge-without-vertexelimination ordering of independent interest. We give several applications of the algorithm, including computing the Hadwiger number of a chordal bipartite graph, solving the unit-time bipartite margin-shop scheduling problem in the case in which the bipartite complement of the precedence graph is chordal bipartite, and determining — in a totally balanced binary matrix — the largest size of a square sub-matrix that is permutation equivalent to a matrix with all zero entries above the main diagonal.