On the spectrum of the forced matching number of 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 originally arose in chemistry in the study of molecular resonance structures. Similar concepts have been studied for block designs and graph colorings under the name defining set, and for Latin squares under the name critical set. Recently several papers have appeared on the study of forcing sets for other graph theoretic concepts such as dominating sets, orientations, and geodetics. Whilst there has been some study of forcing sets of matchings of hexagonal systems in the context of chemistry, only a few other classes of graphs have been considered. Here we study the spectrum of possible forced matching numbers for the grids Pm×Pn, discuss the concept of a forcing set for some other specific classes of graphs, and show that the problem of finding the smallest forcing number of graphs is NP–complete. ∗ Research is partly supported by the Institute for Studies in Theoretical Physics and Mathematics (IPM) 148 P. AFSHANI, H. HATAMI AND E.S. MAHMOODIAN