Localization game for random graphs

Dudek, Andrzej; English, Sean; Frieze, Alan; MacRury, Calum; Prałat, Paweł

doi:10.1016/j.dam.2021.12.002

Cited by 8 publications

(9 citation statements)

References 10 publications

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“…The metric dimension was also studied for the G(n, p) model. The statements of the bounds for β(G) with G ∈ G(n, p) obtained in [2] are slightly technical, but the following observations can be made: for sparser graphs (that is, graphs of diameter at least three a.a.s., which corresponds to i ≥ 2 in the discussion below), it follows from [2] and [10]…”

Section: Related Resultsmentioning

confidence: 99%

“…was studied in [11]. The bounds for dense graphs were consecutively improved in [10], and the arguments were extended to sparse graphs.…”

Section: Related Resultsmentioning

confidence: 99%

“…In particular, the localization game is a combinatorial game and so one of the players must have a winning strategy, that is, a strategy which wins against all of the other player's strategies simultaneously. We direct the reader to [10] for a longer discussion.…”

Section: Reformulation Of the Game With Perfect Information For The Copsmentioning

confidence: 99%

“…Since the associated vertices a have to be not only in R 1 but also at distance at most ε from some vertex b in R 2 , we expose vertices in R 1 and check how many of them are close to some vertex b in R 2 . The number of such vertices a is stochastically bounded from above by the random variable Y ∼ Po(ξ) with ξ := (14πεr/3)(πε 2 ) = 14π 2 ε 3 r/3 = (14π 2 /3) log n > 46 log n. It follows from (10) that with probability 1− o(n −2 ), Y ≤ 2ξ = 28π 2 ε 3 r/3 < 100ε 3 r. Each such vertex a that appears eliminates at most one special pair, and so the desired bound holds, and the proof is finished.…”

Section: Suppose Thatmentioning

confidence: 99%

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Localization Game for Random Geometric Graphs

Lyuben¹,

Mitsche²,

Prałat³

2021

Preprint

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The localization game is a two player combinatorial game played on a graph G = (V, E). The cops choose a set of vertices S 1 ⊆ V with |S 1 | = k. The robber then chooses a vertex v ∈ V whose location is hidden from the cops, but the cops learn the graph distance between the current position of the robber and the vertices in S 1 . If this information is sufficient to locate the robber, the cops win immediately; otherwise the cops choose another set of vertices S 2 ⊆ V with |S 2 | = k, and the robber may move to a neighbouring vertex. The new distances are presented to the robber, and if the cops can deduce the new location of the robber based on all information they accumulated thus far, then they win; otherwise, a new round begins. If the robber has a strategy to avoid being captured, then she wins. The localization number is defined to be the smallest integer k so that the cops win the game. In this paper we determine the localization number (up to poly-logarithmic factors) of the random geometric graph G ∈ G(n, r) slightly above the connectivity threshold.

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Section: Related Resultsmentioning

confidence: 99%

“…was studied in [11]. The bounds for dense graphs were consecutively improved in [10], and the arguments were extended to sparse graphs.…”

Section: Related Resultsmentioning

confidence: 99%

Section: Reformulation Of the Game With Perfect Information For The Copsmentioning

confidence: 99%

Section: Suppose Thatmentioning

confidence: 99%

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Localization Game for Random Geometric Graphs

Lyuben¹,

Mitsche²,

Prałat³

2021

Preprint

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“…The localization number of graph products was considered in [13], and in diameter 2 graphs such as Kneser, Moore, and polarity graphs in [6]. Localization was studied in random binomial graphs in [19,20] and in random geometric graphs in [25].…”

Section: Introductionmentioning

confidence: 99%

The localization capture time of a graph

Behague¹,

Bonato²,

Huggan³

et al. 2021

Preprint

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The localization game is a pursuit-evasion game analogous to Cops and Robbers, where the robber is invisible and the cops send distance probes in an attempt to identify the location of the robber. We present a novel graph parameter called the capture time, which measures how long the localization game lasts assuming optimal play. We conjecture that the capture time is linear in the order of the graph, and show that the conjecture holds for graph families such as trees and interval graphs. We study bounds on the capture time for trees and its monotone property on induced subgraphs of trees and more general graphs. We give upper bounds for the capture time on the incidence graphs of projective planes. We finish with new bounds on the localization number and capture time using treewidth.

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