Domination, eternal domination and clique covering

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

doi:10.7151/dmgt.1799

Cited by 4 publications

(3 citation statements)

References 6 publications

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“…Let G be a triangle-free graph such that γ ∞ (G) < θ(G) = ⌈ n 2 ⌉. Fact 5.4 implies that G ▷◁ K 2 is triangle-free; as a result, θ(G ▷◁ K 2 ) ≥ n. Observe that the vertices of G ▷◁ K 2 can be partitionned into two subsets, each of which induces a subgraph isomorphic to G. This means that [16] posed the following question.…”

Section: Triangle-free Graphsmentioning

confidence: 99%

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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Section: Triangle-free Graphsmentioning

confidence: 99%

Eternal domination and clique covering

MacGillivray

Mynhardt

Virgile

2022

EJGTA

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“…Eternal dominating sets were first defined in [2]. There are many open questions in the field, references [4,6,8,9], and [10] contain a list of some of these. We state here the previously posed question for Cartesian products, which is Question 7.9 in [8] and Problem 8 in [9].…”

Section: Introductionmentioning

confidence: 99%

On eternal domination and Vizing-type inequalities

Driscoll

Klostermeyer

Krop

et al. 2020

AKCE International Journal of Graphs and Combinatorics

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We show sharp Vizing-type inequalities for eternal domination. Namely, we prove that for any graphs G and H, c 1 ðG £ HÞ ! aðGÞc 1 ðHÞ, where c 1 is the eternal domination function, a is the independence number, and £ is the strong product of graphs. This addresses a question of Klostermeyer and Mynhardt. We also show some families of graphs attaining the strict inequality c 1 ðG w HÞ > c 1 ðGÞc 1 ðHÞ where w is the Cartesian product. For the eviction model of eternal domination, we show a sharp upper bound for e 1 ðG £ HÞ:

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“…The product G✷K 2 (K k denotes the complete graph on k vertices) is called the prism of G; one can informally think of the prism of G as the graph obtained by taking two copies of G and adding a matching between corresponding vertices. In [4], the following conjecture is put forward:…”

Section: Introductionmentioning

confidence: 99%

Eternal domination on prisms of graphs

Krim-Yee

Seamone

Virgile

2020

Discrete Applied Mathematics

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An eternal dominating set of a graph G is a set of vertices (or "guards") which dominates G and which can defend any infinite series of vertex attacks, where an attack is defended by moving one guard along an edge from its current position to the attacked vertex. The size of the smallest eternal dominating set is denoted γ ∞ (G) and is called the eternal domination number of G. In this paper, we answer a conjecture of Klostermeyer and Mynhardt [Discussiones Mathematicae Graph Theory, vol. 35,, showing that there exist there are infinitely many graphs G such that γ ∞ (G) = θ(G) and γ ∞ (G✷K 2 ) < θ(G✷K 2 ), where θ(G) denotes the clique cover number of G.

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Domination, eternal domination and clique covering

Cited by 4 publications

References 6 publications

Eternal domination and clique covering

Eternal domination and clique covering

On eternal domination and Vizing-type inequalities

Eternal domination on prisms of graphs

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