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

Abstract

Given a simple connected graph G, the metric dimension dim (Formula presented.) (and edge metric dimension edim (Formula presented.)) is defined as the cardinality of a smallest vertex subset (Formula presented.) 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 (Formula presented.) and edim (Formula presented.) 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.