On the Connected Partition Dimension of a Wheel Related Graph

Let G be a connected graph. For a vertex v ∈ V (G) and an ordered k-partition Π = {S1, S2, ..., Sk} of V (G), the representation of v with respect to Π is the k-vector r(v|Π) = (d(v, S1), d(v, S2), ..., d(v, Sk)), where d(v, Si) denotes the distance between v and Si. 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 < Si > induced by Si (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 J2n be obtained from the wheel with 2n rim vertices W2n by alternately deleting n spokes. In this paper it is shown that for every n ≥ 4 pd(J2n) ≤ 2d √ 2ne + 1 and cpd(J2n) = d(2n + 3)/5e applying Chebyshev’s theorem and an averaging technique.