Further contributions on the outer multiset dimension of graphs

TL;DR

This paper investigates the outer multiset dimension of graphs, characterizing extremal cases, providing polynomial recognition algorithms, and establishing bounds for specific graph products and structures.

Contribution

It offers new characterizations of graphs with maximum and minimum outer multiset dimensions, and analyzes the dimension for graph products and specific graph classes.

Findings

Graphs with maximum outer multiset dimension are regular with diameter at most 2.

Graphs with outer multiset dimension 2 are characterized and recognized efficiently.

The outer multiset dimension of grid graphs P_s square P_t is exactly 3 for s ≥ t ≥ 2.

Abstract

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 $s\ge t\ge 2$.