On some resolving partitions for the lexicographic product of two graphs
Nicolás Campanelli, Ismael G. Yero · International Journal of Computer Mathematics · 2016
Given a connected graph G=(V,E), the distance d(u,v) between two vertices u,v∈V is the length of a shortest u−v path in G. The distance d(v,P) between a vertex v∈V and a subset P⊂V is defined as min{d(v,x):x∈P}. An ordered partition Π={P1,P2,…,Pt} of vertices of G is a resolving partition of G, if for any two different vertices u,v of G there exists Pi∈Π such that d(u,Pi)≠d(v,Pi). The partition dimension of G is the minimum number of sets in any resolving partition of G. In this article, we study the partition dimension of the lexicographic product of two graphs.