Computing the edge metric dimension of convex polytopes related graphs
Muhammad Ahsan, Zohaib Zahid, Sohail Zafar, Arif Rafiq, Muhammad Sarwar Sindhu, Muhammad Awais Umar
Abstract
Let \(G=(V(G),E(G))\) be a connected graph and \(d(f,y)\) denotes the distance between edge \(f\) and vertex \(y\), which is defined as \(d(f,y) = \min \{d(p,y),d(q,y)\}\), where \(f=pq\). A subset \(W_E \subseteq V(G)\) is called an edge metric generator for graph \(G\) if for every two distinct edges \(f_1, f_2 \in E(G)\), there exists a vertex \(y\in W_E\) such that \(d(f_1,y) \neq d(f_2,y)\). An edge metric generator with minimum number of vertices is called an edge metric basis for graph \(G\) and the cardinality of an edge metric basis is called the edge metric dimension represented by \(edim(G)\). In this paper, we study the edge metric dimension of flower graph \({f}_{n\times 3}\) and also calculate the edge metric dimension of the prism related graphs \(D_{n}^{'}\) and \(D_{n}^{t}\). It has been concluded that the edge metric dimension of \(D_{n}^{'}\) is bounded, while of \({f}_{n\times 3}\) and \(D_{n}^{t}\) is unbounded.