On the Constant Edge Resolvability of Some Unicyclic and Multicyclic Graphs

Abstract

Assume that $G=\left(V\left(G\right),E\left(G\right)\right)$ is a connected graph. For a set of vertices $W_E\subseteq V\left(G\right)$ , two edges $g_1,g_2\in E\left(G\right)$ are distinguished by a vertex $x_1\in W_E$ , if $d\left(x_1,g_1\right)\ne d\left(x_1,g_2\right)$ . $W_E$ is termed edge metric generator for $G$ if any vertex of $W_E$ distinguishes every two arbitrarily distinct edges of graph $G$ . Furthermore, the edge metric dimension of $G$ , indicated by $\text{edim}\left(G\right)$ , is the cardinality of the smallest $W_E$ for $G$ . The edge metric dimensions of the dragon, kayak paddle, cycle with chord, generalized prism, and necklace graphs are calculated in this article.