Alliance free and alliance cover sets |
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| Authors: | Juan Alberto Rodriguez-Velazquez José María Sigarreta Ismael Gonzalez Yero Sergio Bermudo |
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| Institution: | 1.Departament d’Enginyeria Informàtica i Matemàtiques,Universitat Rovira i Virgili,Tarragona,Spain;2.Faculty of Mathematics,Autonomous University of Guerrero,Acapulco, Guerrero,Mexico;3.Department of Economy, Quantitative Methods and Economic History,Pablo de Olavide University,Sevilla,Spain |
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| Abstract: | A defensive (offensive) k-alliance in Γ = (V,E) is a set S ⊆ V such that every υ in S (in the boundary of S) has at least k more neighbors in S than it has in V / S. A set X ⊆ V is defensive (offensive) k-alliance free, if for all defensive (offensive) k-alliance S, S/X ≠ ∅, i.e., X does not contain any defensive (offensive) k-alliance as a subset. A set Y ⊆ V is a defensive (offensive) k-alliance cover, if for all defensive (offensive) k-alliance S, S ∩ Y ≠ ∅, i.e., Y contains at least one vertex from each defensive (offensive) k-alliance of Γ. In this paper we show several mathematical properties of defensive (offensive) k-alliance free sets and defensive (offensive) k-alliance cover sets, including tight bounds on their cardinality. |
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| Keywords: | Defensive alliance offensive alliance alliance free set alliance cover set |
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