Computation of Metric Dimension of Certain Subdivided Convex Polytopes

Abstract

The distance $d\left(z_1,z_2\right)$ from vertex $z_1\in V\left(G\right)$ to $z_2\in V\left(G\right)$ is minimum length of $\left(z_1,z_2\right)$ -path in a given connected graph $G$ having E(G) and V(G) edges and vertices’/nodes’ sets, respectively. Suppose $Z=\left\{z_1,z_2,z_3,\dots ,z_m\right\}\subseteq V\left(G\right)$ is an order set and $c\in V\left(G\right)$ , and the code of $c$ with reference to $Z$ is the m-tuple {d(c, z1), d(c, z2), d(c, z13), …, d(c, zk)}. Then, Z is named as the locating set or resolving set if each node of $G$ has unique code. A locating set of least cardinality is described as a basis set for the graph $G$ , and its cardinal number is referred to as metric dimension symbolized by $\dim \left(G\right)$ . Metric dimension of certain subdivided convex polytopes $S{T}_n$ has been computed, and it is concluded that just four vertices are sufficient for unique coding of all nodes belonging to this family of convex polytopes.