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) ≤ 2 ⌈√2n⌉ + 1 and cpd(J 2n) = ⌈(2n + 3)/5⌉ applying Chebyshev's theorem and an averaging technique. © 2012 Springer-Verlag.