Edge resolving number of pentagonal circular ladder

Abstract

Let $G=G(V,E)$ be a non-trivial simple connected graph. The length of the shortest path between two vertices $p$ and $q$, represented by $d(p, q)$, is called the distance between the vertices $p$ and $q$. The distance between an edge $\varepsilon=pq$ and a vertex $r$ in $G$ is defined as $d(\varepsilon, r)=\mbox{min}\{d(p, r), d(q, r)\}$. If $d(r, p)\neq d(r, q)$, then the vertex $r$ is said to distinguish (resolve or recognize) two elements (edges or vertices) $p, q\in V\cup E$. The minimum cardinality of a subset $R$ ($R_e$) of vertices such that all other vertices (edges) of the graph $G$ are uniquely determined by their distances to the vertices in $R$ ($R_e$) is the metric dimension (edge metric dimension) of a graph $G$. In this article, we consider a family of pentagonal circular ladder $(P_m)$ and investigate its edge metric dimension. We show that, for $P_m$ the edge metric dimension is strictly greater than its metric dimension. Additionally, we answer a problem raised in the recent past, regarding the edge metric dimension of a family of a planar graph $R_m$ (exists in the literature).