ON ON THE PARTITION DIMENSION OF A SUBDIVISION OF COMPLETE GRAPH AND COMPLETE BIPARTITE GRAPH - Hasibuan, Ismail Mulia; ABIDIN, WAHYUNI (Departemen Matematika dan Sains Data FMIPA Universitas Andalas Padang, 2025)
Let G(V,E) be connected graph. The distance between two vertices u,vV(G), denoted by d(u,v), is the length of a shortest path from u to v in G. The distance from a vertex vV(G) to a set SV(G) is defined as min{d(v,x)|xS}. The partition Π={S_1,S_2,...,S_k} of V(G) is called a resolving partition of G if the vectors (d(v,S_1),d(v,S_2),...,d(v,S_k)) 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={e_1,e_2,...,e_t}E(G), for some t. The subdivision of a graph G on the edge set A, denoted by S(G(A;n_1,n_2,...,n_t)), is a graph obtained from the graph G by replacing edge e_1 with a path of length n_1+2, edge e_2 with a path of length n_2+2, up to edge e_t with a path of length n_t+2, respectively. In this paper, we determine the partition dimension of S(K_n(A;r_1,r_2, …, r_t)) for n≥ 3 and t≤3. We also derive that partition dimension of S(K_m,n(A;r_1,r_2, …,r_t)) for m≥ n≥ 2 and t≤3