The local metric dimension of subgraph-amalgamation of graphs

TL;DR

This paper investigates the local metric dimension of subgraph-amalgamations of graphs, providing tight bounds especially when the subgraphs are isometric embeddings, advancing understanding of graph distinguishing sets.

Contribution

It introduces tight bounds for the local metric dimension in subgraph-amalgamations, focusing on cases with isometric subgraphs, a novel extension in graph theory.

Findings

Derived tight bounds for local metric dimension

Focused on subgraph-amalgamation with isometric embeddings

Enhanced understanding of vertex distinguishing sets

Abstract

A vertex $v$ is said to distinguish two other vertices $x$ and $y$ of a nontrivial connected graph G if the distance from $v$ to $x$ is different from the distance from $v$ to $y$. A set $S\subseteq V(G)$ is a local metric set for $G$ if every two adjacent vertices of $G$ are distinguished by some vertex of $S$. A local metric set with minimum cardinality is called a local metric basis for $G$ and its cardinality, the local metric dimension of $G$, denoted by $\dim_l(G)$. In this paper we present tight bounds for the local metric dimension of subgraph-amalgamation of graphs with special emphasis in the case of subgraphs which are isometric embeddings.