Optimal Fault-Tolerant Resolving Set of Power Paths

Abstract

In a simple connected undirected graph G, an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v, there is a vertex w∈R such that d(u,w)≠d(v,w). A resolving set F for the graph G is a fault-tolerant resolving set if for each v∈F, F∖{v} is also a resolving set for G. In this article, we determine an optimal fault-resolving set of r-th power of any path Pn when n≥r(r−1)+2. For the other values of n, we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of Pmr from a fault-tolerant resolving set of Pnr where m<n.