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(v,x)|x∈ S\. The partition Π=₁,S₂,...,Sₖ\ of $V(G)$ is called a resolving partition of G if the vectors (d(v,S₁),d(v,S₂),...,d(v,Sₖ)) 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=₁,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(Kₙ(A;r₁,r₂, ⋯, rₜ)) for n≥ 3 and t≤3. We also derive that partition dimension of S(Km,n(A;r₁,r₂, ⋯,rₜ)) for m≥ n≥ 2 and t≤3