On star partition dimension of the generalized gear graph

For a connected graph G and any two vertices u and v in G, let d(u, v) denote the distance between u and v. For a subset S of V (G), the distance between a vertex v and S is d(v, S) = min{d(v, x) | x 2 S}. For an ordered k-partition of V (G) ¦ = {S1, S2, . . . , Sk} and a vertex v, the representation of v with respect to ¦ is the k-vector r(v | ¦) = (d(v, S1), d(v, S2), . . . d(v, Sk)). ¦ is a resolving partition for G if the k-vectors r(v | ¦), v 2 V (G) are distinct. The minimum k for which there exists a resolving k-partition of V (G) is the partition dimension of G, denoted by pd(G). ¦ = {S1, S2, . . . , Sk} is a star resolving k-partition for G if it is a resolving partition and each subgraph induced by Si, 1 · i · k, is a star. The minimum k for which there exists a star resolving k-partition of V (G) is the star partition dimension of G, denoted by spd(G). Let Jk,n be the graph obtained from the wheel Wkn by keeping spokes with step k, for k ¸ 2 and n ¸ 2. In this paper the star partition dimension for this family of graphs is determined.