On the partition dimension of some wheel related graphs

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 \(d(v, S_i) = min_{w∈S_i} d(v, w)(1 ≤ i ≤ k)\). 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 \(pd(G)\). 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 \(k\).