On the Partition Dimension of Cartesian Product Graphs

Let G = (V, E) be a connected graph. The distance between two vertices u, v 2 V, denoted by d(u, v), is the length of a shortest u À v path in G. The distance between a vertex v 2 V and a pd(G), is the minimum number of sets in any resolving partition of G. In this article we study the partition dimension of Cartesian product graphs. More precisely, we show that for all pairs of connected graphs G, H, pd(G Â H) 6 pd(G) + pd(H) and pd(G Â H) 6 pd(G) + dim(H), where dim(H) denotes the metric dimension of H. Consequently, we show that pd(G Â H) 6 dim(G) + dim(H) + 1. The concepts of resolvability and location in graphs were described independently by Harary and Melter [9] and Slater [17], to define the same structure in a graph. After these papers were published several authors developed diverse theoretical works about this topic [2–8,14]. Slater described the usefulness of these ideas into long range aids to navigation [17]. Also, these concepts have some applications in chemistry for representing chemical compounds [12,13] or to problems of pattern recognition and image processing, some of which involve the use of hierarchical data structures [15]. Other applications of this concept to navigation of robots in networks and other areas appear in [5,11,14]. Some variations on resolvability or location have been appearing in the literature, like those about conditional resolvability [16], locating domination [10], resolving domination [1] and resolving partitions [4,7,8]. 1 6 i 6 k, denotes the distance between the vertices v and v i. We say that S is a resolving set of G if for every pair of distinct vertices u, v 2 V, r(ujS) – r(vjS). The metric dimension 1 of G is the minimum cardinality of any resolving set of G, and it is denoted by dim(G). The metric dimension of graphs is studied in [2–6,18].