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 II ={S1; S2; … Sk} of V (G), the representation of v with respect to II is the k-vector r(v|II) = (d(v; S1); d(v; S2); …, d(v; Sk))where d(v; Si) = min w€Si d(v;w)(1 ≤ i ≤ k). The k-partition II is said to be resolving if the k-vectors r(v|II), 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,