Local (Outer) Multiset Dimensions of Graphs

TL;DR

This paper introduces and studies the concepts of local and outer multiset dimensions in graphs, providing properties, bounds, and exact values for certain graph classes, expanding understanding of graph resolving sets.

Contribution

It defines new graph parameters, the local and outer multiset dimensions, and explores their properties, bounds, and specific values for graphs with small diameter.

Findings

Lower bounds for local and outer multiset dimensions.

Necessary conditions for finite local multiset dimension.

Exact dimensions for some small diameter graphs.

Abstract

Let $G$ be a finite, connected, undirected, and simple graph and $W$ be a set of vertices in $G$. A representation multiset of a vertex $u$ in $V(G)$ with respect to $W$ is defined as the multiset of distances between $u$ and the vertices in $W$. If every two adjacent vertices in $V(G)$ have a distinct multiset representation, the set $W$ is called a local multiset resolving set of $G$. If $G$ has a local multiset resolving set, then this set with the smallest cardinality is called the local multiset basis, and its cardinality is the local multiset dimension of $G$. Otherwise, $G$ is said to have an infinite local multiset dimension. On the other hand, if every two adjacent vertices in $V(G) \backslash W$ have a distinct representation multiset, the set $W$ is called a local outer multiset resolving set of $G$. Such a set with the smallest cardinality is called the local outer multiset basis, and its cardinality is the local outer multiset dimension of $G$. Unlike the local multiset dimension, every graph has a finite local outer multiset dimension. This paper presents some basic properties of local (outer) multiset dimensions, including lower bounds for the dimensions and a necessary condition for a finite local multiset dimension. We also determine local (outer) multiset dimensions for some graphs of small diameter.