On Cycle Related Graphs with Constant Metric Dimension

Abstract

Let G be a connected graph and d(x, y) be the distance between the vertices x and y. A subset of vertices W = {w(1), ..., w(k)} is called a resolving set for G if for every two distinct vertices x, y is an element of V(G), there is a vertex w(i) is an element of W such that d(x, w(i)) not equal d(y, w(i)). 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 dim(G). A family of connected graphs G is said to be a family with constant metric dimension if dim(G) is finite and does not depend upon the choice of G in G. In this paper, we show that generalized Petersen graphs P(n, 2), antiprisms A(n) and Harary graphs H-4,H-n for n not equal 1 (mod 4) are families of regular graphs with constant metric dimension and raise some questions in a more general setting.