Resolving sets tolerant to failures in three-dimensional grids

An ordered set S of vertices of a graph G is a resolving set for G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set. In this paper we study resolving sets tolerant to several failures in three-dimensional grids. Concretely, we seek for minimum cardinality sets that are resolving after removing any k vertices from the set. This is equivalent to finding (k+1)-resolving sets, a generalization of resolving sets, where, for every pair of vertices, the vector of distances to the vertices of the set differs in at least k + 1 coordinates. This problem is also related with the study of the (k + 1)-metric dimension of a graph, defined as the minimum cardinality of a (k + 1)-resolving set. In this work, we first prove that the metric dimension of a three-dimensional grid is 3 and establish some properties involving resolving sets in these graphs. Secondly, we determine the values of k >= 1 for which there exists a (k + 1)-resolving set and construct such a resolving set of minimum cardinality in almost all cases.