Identifying the Exact Value of the Metric Dimension and Edge Dimension of Unicyclic Graphs

Given a simple connected graph G, the metric dimension dim$(G)$ (and edge metric dimension edim$(G)$) is defined as the cardinality of a smallest vertex subset $S\subseteq V(G)$ for which every two distinct vertices (and edges) in G have distinct distances to a vertex of S. It is an interesting topic to discuss the relation between these two dimensions for some class of graphs. This paper settles two open problems on this topic for unicyclic graphs. We recently learned that Sedlar and Škrekovski settled these problems, but our work presents the results in a completely different way. By introducing four classes of subgraphs, we characterize the structure of a unicyclic graph G such that dim$(G)$ and edim$(G)$ are equal to the cardinality of any minimum branch-resolving set for unicyclic graphs. This generates an approach to determine the exact value of the metric dimension (and edge metric dimension) for a unicyclic graph.