Fault-Tolerant Metric Dimension of $P(n,2)$ with Prism Graph

Abstract

Let $G$ be a connected graph and $d(a,b)$ be the distance between the vertices $a$ and $b$. A subset $U=\{u_1,u_2,\cdots ,u_k\}$ of the vertices is called a resolving set for $G$ if for every two distinct vertices $a,b\in V(G)$, there is a vertex $u_{\xi }\in U$ such that $d(a,u_{\xi })\ne d(b,u_{\xi })$. A resolving set containing a minimum number of vertices is called a metric basis for $G$ and the number of vertices in a metric basis is its metric dimension denoted by $dim(G)$. A resolving set $U$ for $G$ is fault-tolerant if $U\setminus \{u\}$ is also a resolving set, for each $u\in U$, and the fault-tolerant metric dimension of $G$ is the minimum cardinality of such a set. In this paper we introduce the study of the fault-tolerant metric dimension of $P(n,2)$ with prism graph.

9 pages, 2 figures