On the Geodesic Identification of Vertices in Convex Plane Graphs

Abstract

A shortest path between two vertices $u$ and $v$ in a connected graph $G$ is a $u-v$ geodesic. A vertex $w$ of $G$ performs the geodesic identification for the vertices in a pair $\left(u,v\right)$ if either $v$ belongs to a $u-w$ geodesic or $u$ belongs to a $v-w$ geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in $G$ is called the strong metric dimension of $G$ . In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.