On Some families of Path-related graphs with their edge metric dimension

Abstract

Locating the origin of diffusion in complex networks is an interesting but challenging task. It is crucial for anticipating and constraining the epidemic risks. Source localization has been considered under many feasible models. In this paper, we study the localization problem in some path-related graphs and study the edge metric dimension. A subset $L_E\subseteq V_G$ is known as an edge metric generator for $G$ if, for any two distinct edges $e_1,e_2\in E$, there exists a vertex $a\subseteq L_E$ such that $d(e_1,a)\ne d(e_2,a)$. An edge metric generator that contains the minimum number of vertices is termed an edge metric basis for $G$, and the number of vertices in such a basis is called the edge metric dimension, denoted by $di{m}_e\left(G\right)$. An edge metric generator with the fewest vertices is called an edge metric basis for $G$. The number of vertices in such a basis is the edge metric dimension, represented as $di{m}_e\left(G\right)$. In this paper, the edge metric dimension of some path-related graphs is computed, namely, the middle graph of path $M\left(P_n\right)$ and the splitting graph of path $S\left(P_n\right)$.