Partition dimension of generalized Peterson and Harary graphs

Abstract

The distance of a connected, simple graph PP is denoted by d(α1,α2),d(α1,α2), which is the length of a shortest path between the vertices α1,α2∈V(P),α1,α2∈V(P), where V(P)V(P) is the vertex set of P.P. The ll-ordered partition of V(P)V(P) is K={K1,K2,…,Kl}.K={K1,K2,…,Kl}. A vertex α∈V(P),α∈V(P), and r(α|K)={d(α,K1),d(α,K2),…,d(α,Kl)}r(α|K)={d(α,K1),d(α,K2),…,d(α,Kl)} be a ll-tuple distances, where r(α|K)r(α|K) is the representation of a vertex αα with respect to set K.K. If r(α|K)r(α|K) of αα is unique, for every pair of vertices, then KK is the resolving partition set of V(P).V(P). The minimum number ll in the resolving partition set KK is known as partition dimension (pd(P)pd(P)). In this paper, we studied the generalized families of Peterson graph, Pλ,χPλ,χ and proved that these families have bounded partition dimension.