On the forced matching numbers of bipartite graphs

Let G be a graph that admits a perfect matching. A forcing set for a perfect matching M of G is a subset S of M , such that S is contained in no other perfect matching of G. This notion has arisen in the study of .nding resonance structures of a given molecule in chemistry. Similar concepts have been studied for block designs and graph colorings under the name de/ning set, and for Latin squares under the name critical set. There is some study of forcing sets of hexagonal systems in the context of chemistry, but only a few other classes of graphs have been considered. For the hypercubes Qn, it turns out to be a very interesting notion which includes many challenging problems. In this paper we study the computational complexity of .nding the forcing number of graphs, and we give some results on the possible values of forcing number for di4erent matchings of the hypercube Qn. Also we show an application to critical sets in back circulant Latin rectangles. c © 2003 Elsevier B.V. All rights reserved. MSC: 05C70