Subdivisions in the Robber Locating Game

We consider a game in which a cop searches for a moving robber on a graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any n-vertex graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$ obtained by replacing each edge of $G$ by a path of length $m$, if $m \geqslant n$. They conjectured that this bound was best possible for complete graphs, but the present authors showed that in fact the cop wins on $K^{1/m}$ if and only if $m \geqslant n/2$, for all but a few small values of $n$. In this paper we extend this result to general graphs by proving that the cop has a winning strategy on $G^{1/m}$ provided $m \geqslant n/2$ for all but a few small values of $n$; this bound is best possible. We also consider replacing the edges of $G$ with paths of varying lengths.Comment: 13 Page

Locating a robber with multiple probes

We consider a game in which a cop searches for a moving robber on a connected graph using distance probes, which is a slight variation on one introduced by Seager. Carragher, Choi, Delcourt, Erickson and West showed that for any $n$-vertex graph $G$ there is a winning strategy for the cop on the graph $G^{1/m}$ obtained by replacing each edge of $G$ by a path of length $m$, if $m\geq n$. The present authors showed that, for all but a few small values of $n$, this bound may be improved to $m\geq n/2$, which is best possible. In this paper we consider the natural extension in which the cop probes a set of $k$ vertices, rather than a single vertex, at each turn. We consider the relationship between the value of $k$ required to ensure victory on the original graph and the length of subdivisions required to ensure victory with $k=1$. We give an asymptotically best-possible linear bound in one direction, but show that in the other direction no subexponential bound holds. We also give a bound on the value of $k$ for which the cop has a winning strategy on any (possibly infinite) connected graph of maximum degree $\Delta$, which is best possible up to a factor of $(1-o(1))$.Comment: 16 pages, 2 figures. Updated to show that Theorem 2 also applies to infinite graphs. Accepted for publication in Discrete Mathematic

Localiser une cible dans un graphe

International audienceLe jeu de la localisation d'une cible (invisible et immobile) dans un graphe a été introduit par Seager en 2013. Dans ce jeu, une cible est placée secrètement sur un sommet et, à chaque tour, il est possible d'interroger un sommet et recevoir, comme réponse, la distance exacte entre ce sommet et la cible. L'objectif est de localiser la cible en minimisant le nombre de tours, et ce, quelle que soit sa position. Nous considérons une généralisation de ce jeu où k sommets peuvent être interrogés à chaque tour. Celle-ci est notamment liée à la notion de dimension métrique d'un graphe. Nous étudions aussi la variante où les distances relatives sont données comme réponses, qui généralise la dimension centroïdale des graphes. Pour les deux variantes, nous montrons que localiser la cible en un nombre minimum de tours est NP-complet en général, lorsque k est fixé. Dans le cas des arbres (à n noeuds) et des distances exactes, nous montrons que le problème est NP-complet lorsque k fait partie de l'entrée. Nous donnons cependant une (+1)-approximation de ce problème : nous présentons un algorithme qui, en temps O(n log n) (indépendant de k), calcule une stratégie pour localiser la cible en au plus un tour de plus que l'optimal. Cet algorithme peut aussi être utilisé pour résoudre exactement le problème en temps $n^{O(k)}$

Sequential Metric Dimension

International audienceSeager introduced the following game in 2013. An invisible and immobile target is hidden at some vertex of a graph $G$. Every step, one vertex $v$ of $G$ can be probed which results in the knowledge of the distance between $v$ and the target. The objective of the game is to minimize the number of steps needed to locate the target, wherever it is. 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 by probing at most $k$ vertices per step. We show this problem is NP-complete when $k$ (resp.,) is a fixed parameter. Our main results are for the class of trees where we prove this problem is NP-complete when $k$ and are part of the input but, despite this, we design a polynomial-time (+1)-approximation algorithm in trees which gives a solution using at most one more step than the optimal one. It follows that the Localization Problem is polynomial-time solvable in trees if $k$ is fixed

Sequential Metric Dimension

International audienceIn 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 $k\geq 1$ vertices can be probed at every step. Our game also generalizes the notion of the {\it metric dimension} of a graph.Precisely, given a graph $G$ and two integers $k,\ell \geq 1$, the {\sc Localization} problem asks whether there exists a strategy to locate a target hidden in $G$ in at most $\ell$ steps and probing at most $k$ vertices per step. We first show that, in general, this problem is \textsf{NP}-complete for every fixed $k \geq 1$ (resp., $\ell \geq 1$).We then focus on the class of trees.On the negative side,we prove that the \Localization problem is \textsf{NP}-complete in trees when $k$ and $\ell$ are part of the input. On the positive side, we design a $(+1)$-approximation algorithm for the problem in $n$-node trees, {\it 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 fixed.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 {\it centroidal dimension} of a graph