Partition dimension of disjoint union of complete bipartite graphs - Haryeni, Debi Oktia; Baskoro, Edy Tri; Saputro, Suhadi Wido (Universitas Islam Negeri Raden Intan Lampung, 2021)
For any (not necessary connected) graph G(V,E) and AV(G), the distance of a vertex xV(G) and A is d(x,A)={d(x,a): aA}. A subset of vertices A resolves two vertices x,y 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 xV(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,yV(G) are resolved by A_i for some i[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.