On the Connected Partition Dimension of a Wheel Related Graph

Abstract

Let G be a connected graph. For a vertex v ∈ V(G) and an ordered k-partition Π = {S ₁,S ₂,...,S _k } of V(G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v,S ₁),d(v,S ₂),...,d(v,S _k )), where d(v,S _i ) denotes the distance between v and S _i . The k-partition Π is said to be resolving if the k-vectors r(v|Π), v ∈ V(G), are distinct. The minimum k for which there is a resolving k-partition of V(G) is called the partition dimension of G, denoted by pd(G). If each subgraph < S _i  > induced by S _i (1 ≤ i ≤ k) is required to be connected in G, the corresponding notions are connected resolving k-partition and connected partition dimension of G, denoted by cpd(G). Let the graph J _2n be obtained from the wheel with 2n rim vertices W _2n by alternately deleting n spokes. In this paper it is shown that for every n ≥ 4 $pd(J_{2n})\leq 2\lceil \sqrt{2n}\rceil +1$ and cpd(J _2n ) = ⌈(2n + 3)/5⌉ applying Chebyshev’s theorem and an averaging technique.