Spy-game on graphs: Complexity and simple topologies

Cohen, Nathann; Martins, Nícolas A.; Inerney, Fionn Mc; Nisse, Nicolas; Pérennès, Stéphane; Sampaio, Rudini Menezes

doi:10.1016/j.tcs.2017.11.015

Cited by 11 publications

(5 citation statements)

References 18 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

Section: Related Workmentioning

confidence: 99%

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

2021

Self Cite

View full text Add to dashboard Cite

show abstract

“…In particular, the spy can only attack or move to a vertex at distance at most s from its current position. 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%

“…In Corollary 13, we gave an upper bound for the eternal cop number of trees that we conjecture to be tight. We mention in this context the Spy game [9], where the attacker in this game moves like an agent on the graph much like in Cops and Eternal Robbers. In the Spy game, the guards must maintain that there is at least one guard at distance at most d ≥ 0 from the spy (attacker) at the end of the guards' turn, and the spy moves at speed s ≥ 2.…”

Section: Further Directionsmentioning

confidence: 99%

The game of Cops and Robbers

Bonato¹,

Prałat²

2017

Graph Searching Games and Probabilistic Methods

View full text Add to dashboard Cite

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.

show abstract

“…Combinatorics and graph theory play an important role in studying game theory; see [1,2,3,4,8,9,10,22].…”

Section: Introductionmentioning

confidence: 99%

Ramsey Achievement Games on Graphs : Algorithms and Bounds

Wang¹,

Huang²,

Zhou³

et al. 2023

Preprint

View full text Add to dashboard Cite

In 1982, Harary introduced the concept of Ramsey achievement game on graphs. Given a graph F with no isolated vertices. Consider the following game played on the complete graph K n by two players Alice and Bob. First, Alice colors one of the edges of K n blue, then Bob colors a different edge red, and so on. The first player who can complete the formation of F in his color is the winner. The minimum n for which Alice has a winning strategy is the achievement number of F , denoted by a(F ).If we replace K n in the game by the completed bipartite graph K n,n , we get the bipartite achievement number, denoted by ba(F ). In his seminal paper, Harary proposed an open problem of determining bipartite achievement numbers for trees. In this paper, we correct ba(mK 2 ) = m + 1 to m and disprove ba(K 1,m ) = 2m − 2 from Erickson and Harary, and extend their results on bipartite achievement numbers. We also find the exact values of achievement numbers for matchings, and the exact values or upper and lower bounds of bipartite achievement numbers on matchings, stars, and double stars. Our upper bounds are obtained by deriving efficient winning strategies for Alice.

show abstract

Spy-game on graphs: Complexity and simple topologies

Cited by 11 publications

References 18 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

Ramsey Achievement Games on Graphs : Algorithms and Bounds

Contact Info

Product

Resources

About