The Outer Multiset Dimension of Toroidal Grids

Let $S$ be a set of vertices in a connected graph $G$. A vertex outside $S$ is represented by the multiset of its distances to the vertices of $S$. The outer multiset dimension $\mathrm{odim}(G)$ is the minimum cardinality of an $S$ for which these representations distinguish all vertices outside $S$. We determine $\mathrm{odim}(C_s\mathrm{\square }{C}_t)$ for all $s,t\ge 3$, answering a problem of Klavžar, Kuziak, and Yero. The values range from $3$ to $8$. The proof combines a half-turn argument giving a universal four-landmark lower bound when both factors have length at least four, explicit three- and four-landmark constructions for the infinite families, and exact finite enumeration on the remaining strip. The collision classification behind the infinite four-landmark construction is certified by exact quantifier elimination in linear integer arithmetic; source code and all finite upper certificates accompany the paper.