Resolving sets tolerant to failures in three-dimensional grids

TL;DR

This paper investigates the minimum size of resolving sets in 3D grids that remain effective after multiple vertex failures, introducing the concept of $(k+1)$-resolving sets and establishing their existence and properties.

Contribution

It determines the metric dimension of 3D grids as 3 and characterizes the existence and construction of minimal $(k+1)$-resolving sets for various values of $k$.

Findings

Metric dimension of 3D grid is 3

Existence of $(k+1)$-resolving sets for certain $k$ values

Constructed minimal $(k+1)$-resolving sets in most cases

Abstract

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 differ 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 $k\ge 1$ for which there exists a $(k+1)$-resolving set and construct such a resolving set of minimum cardinality in almost all cases.