Centroidal bases in graphs

Foucaud, Florent; Klasing, Ralf; Slater, Peter J.

doi:10.1002/net.21560

Cited by 9 publications

(10 citation statements)

References 25 publications

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“…The concept of centroidal bases (see eg. [6]) is similar to the one of metric bases. Once again, detecting devices placed at the vertices C = {c 1 , .…”

Section: Introductionmentioning

confidence: 87%

“…If C is a centroidal locating set of minimal cardinality, then it is said to be a centroidal basis of G, and its cardinality is called the centroidal dimension of G, denoted by CD(G). For instance, it is known that, for any nnode graph G with maximum degree at least 2, then (1 + o(1)) ln n ln ln n CD(G) n − 1 [6]. We note that additional information, whether for some checked vertex the distance from the intruder is 0, allows to remove the degree restriction.…”

Section: Introductionmentioning

confidence: 99%

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Centroidal Localization Game

Bosek

Gordinowicz

Grytczuk

et al. 2018

Electron. J. Combin.

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One important problem in a network $G$ is to locate an (invisible) moving entity by using distance-detectors placed at strategical locations in $G$. For instance, the famous metric dimension of a graph $G$ is the minimum number $k$ of detectors placed in some vertices $\{v_1,\cdots,v_k\}$ such that the vector $(d_1,\cdots,d_k)$ of the distances $d(v_i,r)$ between the detectors and the entity's location $r$ allows to uniquely determine $r$ for every $r \in V(G)$. In a more realistic setting, each device does not get the exact distance to the entity's location. Rather, given locating devices placed in $\{v_1,\cdots,v_k\}$, we get only relative distances between the moving entity's location $r$ and the devices (roughly, for every $1\leq i,j\leq k$, it is provided whether $d(v_i,r) >$, $<$, or $=$ to $d(v_j,r)$). The centroidal dimension of a graph $G$ is the minimum number of devices required to locate the entity, in one step, in this setting.In this paper, we consider the natural generalization of the latter problem, where vertices may be probed sequentially (i.e., in several steps) until the moving entity is located. Roughly, at every turn, a set $\{v_1,\cdots,v_k\}$ of vertices are probed and then the relative order of the distances between the vertices $v_i$ and the current location $r$ of the moving entity is given. If it not located, the moving entity may move along one edge. Let $\zeta^* (G)$ be the minimum $k$ such that the entity is eventually located, whatever it does, in the graph $G$. We first prove that $\zeta^* (T)\leq 2$ for every tree $T$ and give an upper bound on $\zeta^*(G\square H)$ for the cartesian product of graphs $G$ and $H$. Our main result is that $\zeta^* (G)\leq 3$ for any outerplanar graph $G$. We then prove that $\zeta^* (G)$ is bounded by the pathwidth of $G$ plus 1 and that the optimization problem of determining $\zeta^* (G)$ is NP-hard in general graphs. Finally, we show that approximating (up to a small constant distance) the location of the robber in the Euclidean plane requires at most two vertices per turn.

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“…The concept of centroidal bases (see eg. [6]) is similar to the one of metric bases. Once again, detecting devices placed at the vertices C = {c 1 , .…”

Section: Introductionmentioning

confidence: 87%

Section: Introductionmentioning

confidence: 99%

Centroidal Localization Game

Bosek

Gordinowicz

Grytczuk

et al. 2018

Electron. J. Combin.

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“…Relative distances and centroidal dimension Foucaud et al dened a variant of resolving sets, called centroidal bases, where the vertices of a graph must be distinguished by their relative distances to the probed vertices [9]. In this setting, given an integer k ≥ 2, probing a set B = {v 1 , .…”

Section: Related Workmentioning

confidence: 99%

“…The set B is a centroidal basis of G if the relative-distance vectors are distinct for every two vertices of G. The centroidal dimension of G, denoted by CD(G), is the minimum size of a centroidal basis of G [9]. Note that CD(G) ≥ 2 unless G has only one vertex, and that CD(G) is well dened since, clearly, V is a centroidal basis of G. The decision problem associated to the centroidal dimension was shown to be NP-complete, and almost tight bounds on the centroidal dimension of paths have been computed (see [9]).…”

Section: Related Workmentioning

confidence: 99%

Sequential Metric Dimension

et al. 2020

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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.

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“…An (ordered) set X = {x 1 , x 2 , ..., x k } ⊆ V (G) is a locating set if for every w ∈ V (G) the ordered k-tuple (d(x 1 , w), d(x 2 , w), ..., d(x k , w)) uniquely determines w. We say that a vertex x resolves vertices u and v if d(x, u) = d(x, v). Then X is locating if for every two vertices u and v at least one x i ∈ X resolves u and v. For the recently introduced centroidal bases described in Foucaud, Klasing and Slater [9] the set of detectors in X provide just an ordering of the relative distances to an intruder vertex, not the exact distances.…”

Section: Introductionmentioning

confidence: 99%

Open-independent, open-locating-dominating sets

Seo

Slater

2017

ejgta

Self Cite

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A distinguishing set for a graph G = (V, E) is a dominating set D, each vertex v ∈ D being the location of some form of a locating device, from which one can detect and precisely identify any given "intruder" vertex in V (G). As with many applications of dominating sets, the set D might be required to have a certain property for D , the subgraph induced by D (such as independence, paired, or connected). Recently the study of independent locating-dominating sets and independent identifying codes was initiated. Here we introduce the property of open-independence for openlocating-dominating sets.

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Centroidal bases in graphs

Cited by 9 publications

References 25 publications

Centroidal Localization Game

Centroidal Localization Game

Sequential Metric Dimension

Open-independent, open-locating-dominating sets

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