The metric dimension for resolving several objects

• 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.

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 ℓ vertices) instead of just one vertex with the aid of the array of distances. The smallest cardinality of a set S resolving at most ℓ vertices is called ℓ-set-metric dimension. We study the problem of the ℓ-set-metric dimension in two infinite classes of graphs, namely, the two dimensional grid graphs and the n-dimensional binary hypercubes.