Global forcing number for maximal matchings

Let M(G) denote the set of all maximal matchings in a simple graph G, and f:M(G)→{0,1}|E(G)| be the characteristic function of maximal matchings of G. Any set S⊆E(G) such that f|S is an injection is called a global forcing set for maximal matchings in G, and the cardinality of smallest such S is called the global forcing number for maximal matchings of G. In this paper we establish sharp lower and upper bounds on this quantity and prove explicit formulas for certain classes of graphs. At the end, we also state some open problems and discuss some further developments.