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Domination, eternal domination and clique covering [PDF]
Discussiones Mathematicae - Graph Theory, 2015 Eternal and m-eternal domination are concerned with using mobile guards to protect a graph against infinite sequences of attacks at vertices. Eternal domination allows one guard to move per attack, whereas more than one guard may move per attack in the m-
William F Klostermeyer +1 more
exaly +7 more sources
On eternal domination and Vizing-type inequalities [PDF]
AKCE International Journal of Graphs and Combinatorics, 2020 We show sharp Vizing-type inequalities for eternal domination. Namely, we prove that for any graphs G and H, where is the eternal domination function, α is the independence number, and is the strong product of graphs.
Elliot Krop +2 more
exaly +6 more sources
Eternal Domination in Grids [PDF]
Lecture Notes in Computer Science, 2019 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 $\gamma^{\infty}_{all}$ of a graph which is the minimum number of ...
Fionn Mc Inerney, Nicolas Nisse
exaly +10 more sources
Eternal Domination: Criticality and Reachability [PDF]
Discussiones Mathematicae Graph Theory, 2017 We show that for every minimum eternal dominating set, D, of a graph G and every vertex v ∈ D, there is a sequence of attacks at the vertices of G which can be defended in such a way that an eternal dominating set not containing v is reached.
Klostermeyer William F. +1 more
doaj +7 more sources
Eternal Domination of Generalized Petersen Graph [PDF]
Journal of Applied Mathematics, 2021 An eternal dominating set of a graph G is a set of guards distributed on the vertices of a dominating set so that each vertex can be occupied by one guard only.
Ramy Shaheen, Ali Kassem
doaj +5 more sources
Eternal domination and clique covering [PDF]
Electronic Journal of Graph Theory and Applications, 2022 We study the relationship between the eternal domination number of a graph and its clique cove-ring number using both large-scale computation and analytic methods. In doing so, we answer two open questions of Klostermeyer and Mynhardt.
Gary MacGillivray +2 more
doaj +13 more sources
An Eternal Domination Problem in Grids
Theory and Applications of Graphs, 2017 A dynamic domination problem in graphs is considered in which an infinite sequence of attacks occur at vertices with mobile guards; the guard at the attacked vertex is required to vacate the vertex by moving to a neighboring vertex with no guard.
Margaret-Ellen Messinger
exaly +6 more sources
A method for eternally dominating strong grids [PDF]
Discrete Mathematics & Theoretical Computer Science, 2020 In the eternal domination game, 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 and no more than one guard may occupy a vertex.
Alizée Gagnon +7 more
doaj +4 more sources
Eternal Distance-k Domination on Graphs
La Matematica, 2023 Eternal domination is a dynamic process by which a graph is protected from an infinite sequence of vertex intrusions. In eternal distance-$k$ domination, guards initially occupy the vertices of a distance-$k$ dominating set. After a vertex is attacked, guards ``defend'' by each moving up to distance $k$ to form a distance-$k$ dominating set, such that ...
Erin Meger
exaly +4 more sources
Domination and Eternal Domination of Jahangir Graph [PDF]
Open Journal of Discrete Mathematics, 2019 In the eternal dominating set problem, guards form a dominating set on a graph and at each step, a vertex is attacked. We consider the “all guards move” of the eternal dominating set problem. In which one guard has to move to the attacked vertex and all the remaining guards are allowed to move to an adjacent vertex or stay in their current position ...
Ramy Shaheen, Mohammad Assaad
exaly +2 more sources