Study of a Combinatorial Game in Graphs Through Linear Programming

Cohen, Nathann; Inerney, Fionn Mc; Nisse, Nicolas; Pérennès, Stéphane

doi:10.1007/s00453-018-0503-9

Cited by 10 publications

(6 citation statements)

References 17 publications

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“…The eviction model of eternal domination was studied in [16], where a vertex containing a guard is attacked each turn, which forces the guard to move to an adjacent empty vertex with the condition that the guards must maintain a dominating set each turn. The authors of the current paper studied a generalisation of eternal domination, called the Spy game, in [7,8]. For more information and results on the original eternal domination game and its variants, see the survey [19].…”

Section: Related Workmentioning

confidence: 99%

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

2021

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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 first 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. Eternally dominating large grids. Theoretical Computer Science, 794:27-46, 2019]. 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). We then generalise our technique to prove that γ ∞ all (G) = γ(G) + o(γ(G)) for all graphs G ∈ F, where F is a large family of D-dimensional grids which are supergraphs of the D-dimensional Cartesian grid and subgraphs of the Ddimensional strong grid. In particular, F includes both the D-dimensional Cartesian grid and the D-dimensional strong grid.

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Section: Related Workmentioning

confidence: 99%

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

2021

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“…In [9], the game was shown to be NP-hard, paths were resolved, and almost tight bounds were given for cycles. In [10], a polynomial-time algorithm for trees was obtained using Linear Programming and a fractional relaxation of the game, and some asymptotic bounds were obtained for grids.…”

Section: Further Directionsmentioning

confidence: 99%

“…For the Spy Game on trees, a fractional variant and Linear Program gave a polynomialtime algorithm. In that case, there exists an optimal strategy in trees in which there is a unique configuration for the guards for each position of the spy [10]. Cops and Eternal Robbers is different, however, since the robber chooses his position at the beginning of each play.…”

Section: Further Directionsmentioning

confidence: 99%

The game of Cops and Robbers

Bonato¹,

Prałat²

2017

Graph Searching Games and Probabilistic Methods

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We introduce the game of Cops and Eternal Robbers played on graphs, where there are infinitely many robbers that appear sequentially over distinct plays of the game. A positive integer t is fixed, and the cops are required to capture the robber in at most t time-steps in each play. The associated optimization parameter is the eternal cop number, denoted by c ∞ t , which equals the eternal domination number in the case t = 1, and the cop number for sufficiently large t. We study the complexity of Cops and Eternal Robbers, and show that game is NP-hard when t is a fixed constant and EXPTIME-complete for large values of t. We determine precise values of c ∞ t for paths and cycles. The eternal cop number is studied for retracts, and this approach is applied to give bounds for trees, as well as for strong and Cartesian grids.

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“…The authors of the current paper studied a generalization of eternal domination, called the Spy game, in [7,8]. For more information and results on the original eternal domination game and its variants, see the survey [17].…”

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confidence: 99%

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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Study of a Combinatorial Game in Graphs Through Linear Programming

Cited by 10 publications

References 17 publications

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

Eternal Domination: D-Dimensional Cartesian and Strong Grids and Everything in Between

The game of Cops and Robbers

Eternal Domination in Grids

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