Global Forcing Number for Maximal Matchings under Graph Operations

Let $S= \{e_1,\,e_2‎, ‎\ldots,\,e_m\}$ be an ordered subset of edges of a connected graph $G$‎. ‎The edge $S$-representation of an edge set $M\subseteq E(G)$ with respect to $S$ is the‎ ‎vector $r_e(M|S) = (d_1,\,d_2,\ldots,\,d_m)$‎, ‎where $d_i=1$ if $e_i\in M$ and $d_i=0$‎ ‎otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $S$ is a global forcing set for maximal matchings of $G$‎ ‎if $r_e(M_1|S)\neq r_e(M_2|S)$ for any two maximal matchings $M_1$ and $M_2$ 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 $\varphi_{gm}$‎, ‎is the‎ ‎global forcing number (GFN for short) for maximal matchings‎. ‎Similarly‎, ‎for an ordered subset $F = \{v_1,\,v_2‎, ‎\ldots,\,v_k\}$ of $V(G)$‎, ‎the $F$-representation of a vertex set $I\subseteq V(G)$ with respect to $F$ is the‎ ‎vector $r(I|F) = (d_1,\,d_2,\ldots,\,d_k)$‎, ‎where $d_i=1$ if $v_i\in I$ and‎ ‎$d_i=0$ otherwise‎, ‎for each $i\in\{1,\ldots‎ , ‎k\}$‎. ‎We say $F$ is a global forcing set for independent dominatings of $G$‎ ‎if $r(D_1|F)\neq r(D_2|F)$ for any two maximal independent dominating sets $D_1$ and $D_2$ 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 $\varphi_{gi}$‎, ‎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 $\varphi_{gm}$ of some well-known graphs.