Eternal and Secure Domination in Graphs

Klostermeyer, William F.; Mynhardt, C. M.

doi:10.1007/978-3-030-51117-3_13

Cited by 4 publications

(9 citation statements)

References 46 publications

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“…The complement of the Grötzsch graph is the unique smallest graph with independence number 2 and clique covering number at least 4. [17,18] posed the following questions.…”

Section: Corollary 32 ([10]) For Any Integermentioning

confidence: 99%

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Eternal domination and clique covering

MacGillivray

Mynhardt

Virgile

2022

EJGTA

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We study the relationship between the eternal domination number of a graph and its clique covering number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt. We show that the smallest graph having its eternal domination number less than its clique covering number has 10 vertices. We determine the complete set of 10-vertex and 11-vertex graphs having eternal domination numbers less than their clique covering numbers. We show that the smallest triangle-free graph with this property has order 13, as does the smallest circulant graph. We describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. We also consider planar graphs and cubic graphs. Finally, we show that for any integer k ≥ 2 there exist infinitely many graphs having domination number and eternal domination number equal to k containing dominating sets which are not eternal dominating sets.

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“…The complement of the Grötzsch graph is the unique smallest graph with independence number 2 and clique covering number at least 4. [17,18] posed the following questions.…”

Section: Corollary 32 ([10]) For Any Integermentioning

confidence: 99%

“…In other words, the attacker wins if at some time t there is no guard in the neighbourhood of some vertex. The eternal domination number of a graph G, denoted by γ ∞ (G), is the minimum number of guards necessary to respond to any sequence of attacks on G. For a survey on the eternal domination game and its variants, see [17,18].…”

Section: Introductionmentioning

confidence: 99%

Eternal domination and clique covering

MacGillivray

Mynhardt

Virgile

2022

EJGTA

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“…Recently, the eternal domination game and a variant have been studied in digraphs, including orientations of grids and toroidal strong grids [2]. Eternal total domination was studied in [18], where a total dominating set must be maintained by the guards each turn. 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.…”

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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“…As we have seen above, the guards must always be located on the vertices of a dominating set in a graph G in order to defend a sequence of attacks on G. With this in mind, many researchers showed interest in characterizing graphs for which the domination number is equal to the eternal domination number. Klostermeyer and Mynhardt [18] posed the following question.…”

Section: Domination Eternal Domination and Clique Coveringmentioning

confidence: 99%

“…the attacker selects a vertex v on which there is no guard; we say the attacker attacks v. The defender responds by moving a guard on a neighbour of v to v; we say the defender defends v. The guards (or the defenders) win if they are able to respond to the sequence of attacks; otherwise the attacker wins, that is, if at some time t there is no guard in the neighbourhood of the vertex selected by the attacker. The eternal domination number of a graph G, denoted by γ ∞ (G), is the minimum number of guards necessary to respond to any sequence of attacks on G. For a survey on the eternal domination game and its variants, see [17] or [18].…”

Section: Introductionmentioning

confidence: 99%

Eternal Domination and Clique Covering

MacGillivray¹,

Mynhardt²,

Virgile³

2021

Preprint

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We study the relationship between the eternal domination number of a graph and its clique covering number. Using computational methods, we show that the smallest graph having its eternal domination number less than its clique covering number has 10 vertices. This answers a question of Klostermeyer and Mynhardt [Protecting a graph with mobile guards, Appl. Anal. Discrete Math. 10 (2016), no. 1, 1 − 29]. We also determine the complete set of 10-vertex and 11-vertex graphs having eternal domination numbers less than their clique covering numbers.In addition, we study the problem on triangle-free graphs, circulant graphs, planar graphs and cubic graphs. Our computations show that all triangle-free graphs and all circulant graphs of order 12 or less have eternal domination numbers equal to their clique covering numbers, and exhibit 13 triangle-free graphs and 2 circulant graphs of order 13 which do not have this property. Using these graphs, we describe a method to generate an infinite family of triangle-free graphs and an infinite family of circulant graphs with eternal domination numbers less than their clique covering numbers. Our computations also show that all planar graphs of order 11 or less, all 3-connected planar graphs of order 13 or less and all cubic graphs of order less than 18 have eternal domination numbers equal to their clique covering numbers. Finally, we show that for any integer k ≥ 2 there exist infinitely many graphs having domination number and eternal domination number equal to k containing dominating sets which are not eternal dominating sets. This answers another question of Klostermeyer and Mynhardt [Eternal and Secure Domination in Graphs, Topics in domination in graphs, Dev. Math. 64 (2020), 445-478, Springer, Cham].

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Eternal and Secure Domination in Graphs

Cited by 4 publications

References 46 publications

Eternal domination and clique covering

Eternal domination and clique covering

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

Eternal Domination and Clique Covering

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