Perpetually Dominating Large Grids

Lamprou, Ioannis; Martin, Russell; Schewe, Sven

doi:10.1007/978-3-319-57586-5_33

Cited by 2 publications

(5 citation statements)

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“…The best known lower bound for γ ∞ all (P n P m ) for values of n and m large enough, is the domination number with the latter only being recently determined in [13]. The best known upper bound for γ ∞ all (P n P m ) was determined recently in [18], where it was shown that γ ∞ all (P n P m ) = γ(P n P m ) + O(n + m). Note that all the results discussed in this subsection also hold for γ * ∞…”

Section: Related Workmentioning

confidence: 99%

“…The corresponding eternal domination number for this variant will be denoted by γ * ∞ all . This variant is also considered in, e.g., [5,15,18].…”

Section: At Most One Guard At Each Vertexmentioning

confidence: 99%

“…Particular graph classes have also been studied such as paths and cycles [11], trees [15], and proper interval graphs [5]. In particular, the class of grids and graph products has been widely studied [4,10,12,17,18,19,23].…”

Section: Introductionmentioning

confidence: 99%

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Eternal Domination in Grids

Inerney

Nisse

Pérennès

2019

Lecture Notes in Computer Science

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In the eternal domination game played on graphs, an attacker attacks a vertex at each turn and a team of guards must move a guard to the attacked vertex to defend it. The guards may only move to adjacent vertices on their turn. The goal is to determine the eternal domination number γ ∞ all of a graph which is the minimum number of guards required to defend against an infinite sequence of attacks. This paper continues the study of the eternal domination game on strong grids P n P m. Cartesian grids P n P m have been vastly studied with tight bounds existing for small grids such as k × n grids for k ∈ {2, 3, 4, 5}. It was recently proven that γ ∞ all (P n P m) = γ(P n P m) + O(n+m) where γ(P n P m) is the domination number of P n P m which lower bounds the eternal domination number [Lamprou et al., CIAC 2017]. We prove that, for all n, m ∈ N * such that m ≥ n, n 3 m 3 + Ω(n + m) = γ ∞ all (P n P m) = n 3 m 3 + O(m √ n) (note that n 3 m 3 is the domination number of P n P m). Our technique may be applied to other "grid-like" graphs.

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Section: Related Workmentioning

confidence: 99%

“…The corresponding eternal domination number for this variant will be denoted by γ * ∞ all . This variant is also considered in, e.g., [5,15,18].…”

Section: At Most One Guard At Each Vertexmentioning

confidence: 99%

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Eternal Domination in Grids

Inerney

Nisse

Pérennès

2019

Lecture Notes in Computer Science

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“…For instance, the robber may be faster than the cops [2,12], the cops may capture at some distance [4], the surveyed area may be forced to be connected [13], etc. Another approach may be to restrict the games to particular graph classes such as trees [19], grids [16,21], planar graphs [1], bounded treewidth graphs [17,20], etc.…”

Section: Introductionmentioning

confidence: 99%

“…In grids, only a few cases are known: for instance, tight bounds are known in m×n grids for n ≤ 4 [3,10] and the case n = 5 is considered in [26]. The best known general upper bound in grids is nm 5 +O(n+m) [21]. Note that the minimum size of a dominating set in any grid has only been characterized recently [16].…”

mentioning

confidence: 99%

Study of a Combinatorial Game in Graphs Through Linear Programming

et al. 2018

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In the Spy Game played on a graph G, a single spy travels the vertices of G at speed s, while multiple slow guards strive to have, at all times, one of them within distance d of that spy. In order to determine the smallest number of guards necessary for this task, we analyze the game through a Linear Programming formulation and the fractional strategies it yields for the guards. We then show the equivalence of fractional and integral strategies in trees. This allows us to design a polynomial-time algorithm for computing an optimal strategy in this class of graphs. Using duality in Linear Programming, we also provide non-trivial bounds on the fractional guard-number of grids and torus which gives a lower bound for the integral guardnumber in these graphs. We believe that the approach using fractional relaxation and Linear Programming is promising to obtain new results in the field of combinatorial games.

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Perpetually Dominating Large Grids

Cited by 2 publications

References 11 publications

Eternal Domination in Grids

Eternal Domination in Grids

Study of a Combinatorial Game in Graphs Through Linear Programming

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