Smallest Maximal Matchings of Graphs

‎Let $G=(V_G‎, ‎E_G)$ be a simple and connected graph‎. ‎A set $M\subseteq E_G$ is called a matching of‎ 
‎$G$ if no two edges of $M$ are adjacent‎. ‎The number of edges in $M$ is called‎ 
‎its size‎. ‎A matching $M$ is maximal if it cannot be extended to a larger‎ 
‎matching in $G$‎. ‎The smallest size of a maximal matching is called the‎ 
‎saturation number of $G$‎. ‎In this paper we are concerned with the saturation‎ 
‎numbers of lexicographic product of graphs‎. ‎We also address and solve an open‎ 
‎problem about the size of maximum matchings in graphs with a given maximum‎ 
‎degree $\Delta$‎.