Further Contributions on the Outer Multiset Dimension of Graphs

The outer multiset dimension $$\textrm{dim}_\textrm{ms}(G)$$ of a graph G is the cardinality smallest set vertices that uniquely recognize all outside this by using multisets distances to set. It proved $$\textrm{dim}_\textrm{ms}(G) = n(G) - 1$$ if and only regular with diameter at most 2. Graphs $$\textrm{dim}_\textrm{ms}(G)=2$$ are described recognized in polynomial time. A lower bound on lexicographic product H when complete or edgeless, extremal graphs determined. $$\textrm{dim}_\textrm{ms}(P_s\,\square \, P_t) 3$$ for $$s\ge t\ge 2$$ .