Matching preclusion for $n$-grid graphs

A matching preclusion set of a graph is an edge set whose deletion results in a graph without perfect matching or almost perfect matching. The Cartesian product of $n$ paths is called an $n$-grid graph. In this paper, we study the matching preclusion problems for $n$-grid graphs and obtain the following results. If an $n$-grid graph has an even order, then it has the matching preclusion number $n$, and every optimal matching preclusion set is trivial. If the $n$-grid graph has an odd order, then it has the matching preclusion number $n+1$, and all the optimal matching preclusion sets are characterized.