Global Forcing Number for Maximal Matchings under Graph Operations

Let S = {e1, e2, . . . , em} be an ordered subset of edges of a connected graph G. The edge S-representation of an edge set M ⊆ E(G) with respect to S is the vector re(M|S) = (d1, d2, . . . , dm), where di = 1 if ei ∈ M and di = 0 otherwise, for each i ∈ {1, . . . , k}. We say S is a global forcing set for maximal matchings of G if re(M1|S) ̸= re(M2|S) for any two maximal matchings M1 and M2 of G. A global forcing set for maximal matchings of G with minimum cardinality is called a minimum global forcing set for maximal matchings, and its cardinality, denoted by φgm, is the global forcing number (GFN for short) for maximal matchings. Similarly, for an ordered subset F = {v1, v2, . . . , vk} of V (G), the F-representation of a vertex set I ⊆ V (G) with respect to F is the vector r(I|F) = (d1, d2, . . . , dk), where di = 1 if vi ∈ I and di = 0 otherwise, for each i ∈ {1, . . . , k}. We say F is a global forcing set for independent dominatings of G if r(D1|F) ̸= r(D2|F) for any two maximal independent dominating sets D1 and D2 of G. A global forcing set for independent dominatings of G with minimum cardinality is called a minimum global forcing set for independent dominatings, and its cardinality, denoted by φgi, is the GFN for independent dominatings. In this paper we study the GFN for maximal matchings under several types of graph products. Also, we present some upper bounds for this invariant. Moreover, we present some bounds for φgm of some well-known graphs.