Metric dimension and edge metric dimension of unicyclic graphs

TL;DR

This paper determines the exact metric and edge metric dimensions of all unicyclic graphs by characterizing their structure and resolving two open problems in the field.

Contribution

It introduces four subgraph classes to characterize unicyclic graphs with minimal metric and edge metric dimensions, solving open problems in the area.

Findings

Exact values of dim(G) for all unicyclic graphs

Exact values of edim(G) for all unicyclic graphs

Structural characterization of unicyclic graphs with minimal dimensions

Abstract

The metric (resp. edge metric) dimension of a simple connected graph $G$, denoted by dim$(G)$ (resp. edim$(G)$), is the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (resp. edges) in $G$ have distinct distances to a vertex of $S$. It is an interesting topic to discuss the relation between dim$(G)$ and edim$(G)$ for some class of graphs $G$. In this paper, we settle two open problems on this topic for a widely studied class of graphs, called unicyclic graphs. Specifically, we introduce four classes of subgraphs to characterize the structure of a unicyclic graph whose metric (resp . edge metric) dimension is equal to the lower bound on this invariant for unicyclic graphs. Based on this, we determine the exact values of dim$(G)$ and edim$(G)$ for all unicyclic graphs $G$.