ON ON THE PARTITION DIMENSION OF A SUBDIVISION OF COMPLETE GRAPH AND COMPLETE BIPARTITE GRAPH - Hasibuan, Ismail Mulia; ABIDIN, WAHYUNI (Jurnal Matematika UNAND, 2025)
Let $G(V,E)$ be connected graph. The distance between two vertices $u,v\in V(G),$ denoted by $d(u,v),$ is the length of a shortest path from $u$ to $v$ in $G.$ The distance from a vertex $v\in V(G)$ to a set $S\subset V(G)$ is defined as $min\{d(v,x)|x\in S\}$. 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 $v \in V(G)$ are distinct. The partition dimension of $G$, denoted by pd(G), is the smallest $k$ such that $G$ has a resolving $k$-partition. Let $A=\{e_1,e_2,...,e_t\}\subseteq E(G),$ 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 $t≤3$