Further Contributions on the Outer Multiset Dimension of Graphs

Abstract

The outer multiset dimension \(\textrm{dim}_\textrm{ms}(G)\) 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 \(\textrm{dim}_\textrm{ms}(G) = n(G) - 1\) if and only if G is a regular graph with diameter at most 2. Graphs G with \(\textrm{dim}_\textrm{ms}(G)=2\) 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 \(\textrm{dim}_\textrm{ms}(P_s\,\square \, P_t) = 3\) for \(s\ge t\ge 2\).