On on the Partition Dimension of a Subdivision of Complete Graph and Complete Bipartite Graph

Let G (V, E) be connected graph. The distance between two vertices u, v V (G), denoted by d (u, v), is the length of a shortest path from u to v in G. The distance from a vertex v V (G) to a set S V (G) is defined as min\d (v, x) |x S\. The partition =\S₁, S₂,. . . , S₊\ of V (G) is called a resolving partition of G if the vectors (d (v, S₁), d (v, S₂),. . . , d (v, S₊) ) for all v V (G) are distinct. The partition dimension of G, denoted by pd (G), is the smallest k such that G has a resolving k-partition. Let A=\e₁, e₂,. . . , eₜ\ E (G), for some t. The subdivision of a graph G on the edge set A, denoted by S (G (A;n₁, n₂,. . . , nₜ) ), is a graph obtained from the graph G by replacing edge e₁ with a path of length n₁+2, edge e₂ with a path of length n₂+2, up to edge eₜ with a path of length nₜ+2, respectively. In this paper, we determine the partition dimension of S (K₍ (A;r₁, r₂, , rₓ) ) for n 3 and t3. We also derive that partition dimension of S (K₌, ₍ (A;r₁, r₂, , rₓ) ) for m n 2 and t3