On the partition dimension of some wheel related graphs

Abstract

Let G be a connected graph. For a vertex v∈V(G)v∈V(G) and an ordered k−k−partition Π=S1,S2,…,SkΠ=S1,S2,…,Sk of V(G)V(G), the representation of vv with respect to ΠΠ is the k−k−vector r (v|Π)=(d(v,S1),d(v,S2),…,d(v,Sk))(v|Π)=(d(v,S1),d(v,S2),…,d(v,Sk)) where d(v,Si)=minw∈Sid(v,w)(1≤i≤k)d(v,Si)=minw∈Sid(v,w)(1≤i≤k). The k-partition ΠΠ is said to be resolving if the k-vectors r(v|Π),v∈V(G)r(v|Π),v∈V(G), are distinct. The minimum kk for which there is a resolving kk-partition of V(G)V(G) is called the partition dimension of GG, denoted by pd(G)pd(G). In this paper, we give upper bounds for the cardinality of vertices in some wheel related graphs namely gear graph, helm, sunflower and friendship graph with given partition dimension kk.