On the partition dimension and connected partition dimension of wheels

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)). 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). A resolving k-partition Π = {S1, S2, ..., Sk} of V (G) is said to be connected if each subgraph < Si > induced by Si (1 ≤ i ≤ k) is connected in G. The minimum k for which there is a connected resolving k-partition of V (G) is called the connected partition dimension of G, denoted by cpd(G). In this paper, the partition dimension as well as the connected partition dimension of the wheel Wn with n spokes are considered, by showing that d(2n)e ≤ pd(Wn) ≤ 2dne + 1 and cpd(Wn) = d(n + 2)/3e for ∗The research was done while the third author was visiting School of Mathematical Sciences, GC University, Lahore, Pakistan. Research partially supported by the School of Mathematical Sciences, Lahore and by the Higher Education Commission of Pakistan.