Partition dimension of disjoint union of complete bipartite graphs - Haryeni, Debi Oktia; Baskoro, Edy Tri; Saputro, Suhadi Wido (Desimal: Jurnal Matematika, 2021)
For any (not necessary connected) graph $G(V,E)$ and $A\subseteq V(G)$, the distance of a vertex $x\in V(G)$ and $A$ is $d(x,A)=\min\{d(x,a): a\in A\}$. A subset of vertices $A$ resolves two vertices $x,y \in V(G)$ if $d(x,A)≠ d(y,A)$. For an ordered partition $Λ=\{A_1, A_2,…, A_k\}$ of $V(G)$, if all $d(x,A_i)<∞$ for all $x\in V(G)$, then the representation of $x$ under $Λ$ is $r(x|Λ)=(d(x,A_1), d(x,A_2), …, d(x,A_k))$. Such a partition $Λ$ is a resolving partition of $G$ if every two distinct vertices $x,y\in V(G)$ are resolved by $A_i$ for some $i\in [1,k]$. The smallest cardinality of a resolving partition $Λ$ is called a partition dimension of $G$ and denoted by $pd(G)$ or $pdd(G)$ for connected or disconnected $G$, respectively. If $G$ have no resolving partition, then $pdd(G)=∞$. In this paper, we studied the partition dimension of disjoint union of complete bipartite graph, namely $tK_{m,n}$ where $t≥ 1$ and $m≥ n≥ 2$. We gave the necessary condition such that the partition dimension of $tK_{m,n}$ are finite. Furthermore, we also derived the necessary and sufficient conditions such that $pdd(tK_{m,n})$ is either equal to $m$ or $m+1$.