The metric dimension for resolving several objects

A set of vertices S is a resolving set in a graph if each vertex has a unique array of distances to the vertices of S. The natural problem of finding the smallest cardinality of a resolving set in a graph has been widely studied over the years. In this paper, we wish to resolve a set of vertices (up to ź vertices) instead of just one vertex with the aid of the array of distances. The smallest cardinality of a set S resolving at most ź vertices is called ź-set-metric dimension. We study the problem of the ź-set-metric dimension in two infinite classes of graphs, namely, the two dimensional grid graphs and the n-dimensional binary hypercubes. We introduce a way to locate several intruders instead of only one of a resolving set.This prevents mistakes in the location procedure, see Example 2(i).New geometric approach compared to the usual resolving sets (Section 2).We give the complete and optimal results for the two dimensional grid graphs.Optimal results for a very high number of intruders in binary hypercubes.