Sequential Metric Dimension

Abstract

In the localization game, introduced by Seager in 2013, an invisible and immobile target is hidden at some vertex of a graph G. At every step, one vertex v of G can be probed which results in the knowledge of the distance between v and the secret location of the target. The objective of the game is to minimize the number of steps needed to locate the target whatever be its location. We address the generalization of this game where k ≥ 1 vertices can be probed at every step. Our game also generalizes the notion of the metric dimension of a graph. Precisely, given a graph G and two integers k, ≥ 1, the Localization problem asks whether there exists a strategy to locate a target hidden in G in at most steps and probing at most k vertices per step. We rst show that, in general, this problem is NP-complete for every xed k ≥ 1 (resp., ≥ 1). We then focus on the class of trees. On the negative side, we prove that the Localization problem is NPcomplete in trees when k and are part of the input. On the positive side, we design a (+1)-approximation algorithm for the problem in n-node trees, i.e., an algorithm that computes in time O(n log n) (independent of k) a strategy to locate the target in at most one more step than an optimal strategy. This algorithm can be used to solve the Localization problem in trees in polynomial time if k is xed. We also consider some of these questions in the context where, upon probing the vertices, the relative distances to the target are retrieved. This variant of the problem generalizes the notion of the centroidal dimension of a graph.

Metric Dimension Parameterized by Feedback Vertex Set and Other Structural Parameters

“…The metric dimension has also been considered in digraphs, with Bensmail et al [6] providing a summary of the related work in this area. Interestingly, there are many game-theoretic variants of the metric dimension, such as sequential metric dimension [5], the localization game [9,24], and the centroidal localization game [8]. The metric dimension and its variants have been studied for both their theoretical interest and their numerous applications such as in network verification [2], fault-detection in networks [36], pattern recognition and image processing [30], graph isomorphism testing [1], chemistry [11,26], and genomics [35].…”