# Distance-based vertex identification in graphs: the outer multiset dimension

## TL;DR

This paper introduces the concept of outer multiset dimension in graphs, analyzing its properties, calculating exact values for specific graph families, and proving its NP-hardness for general graphs.

## Contribution

It defines the outer multiset dimension, explores its behavior, computes exact values for certain graphs, and establishes NP-hardness for the general problem.

## Key findings

- Exact values for several graph families
- NP-hardness of computing the outer multiset dimension
- Methods for specific cases

## Abstract

Given a graph $G$ and a subset of vertices $S = \{w_1, \ldots, w_t\} \subseteq V(G)$, the multiset representation of a vertex $u\in V(G)$ with respect to $S$ is the multiset $m(u|S) = \{| d_G(u, w_1), \ldots, d_G(u, w_t) |\}$. A subset of vertices $S$ such that $m(u|S) = m(v|S) \iff u = v$ for every $u, v \in V(G) \setminus S$ is said to be a multiset resolving set, and the cardinality of the smallest such set is the outer multiset dimension. We study the general behaviour of the outer multiset dimension, and determine its exact value for several graph families. We also show that computing the outer multiset dimension of arbitrary graphs is NP-hard, and provide methods for efficiently handling particular cases.