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Further contributions on the outer multiset dimension of graphs

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