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Alliance Free and Alliance Cover Sets

Alliance Free and Alliance Cover Sets
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摘要 A defensive （offensive） k-alliance in F = （V, E） is a set S C V such that every v in S （in the boundary of S） has at least k more neighbors in S than it has in V ／ S. A set X C_ V is defensive （offensive） k-alliance free, if for all defensive （offensive） k-alliance S, S／ X ≠ 0, i.e., X does not contain any defensive （offensive） k-alliance as a subset. A set Y C V is a defensive （offensive） k-alliance cover, if for all defensive （offensive） k-alliance S, S ∩ Y ≠ 0, i.e., Y contains at least one vertex from each defensive （offensive） k-alliance of F. 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. A defensive （offensive） k-alliance in F = （V, E） is a set S C V such that every v in S （in the boundary of S） has at least k more neighbors in S than it has in V ／ S. A set X C_ V is defensive （offensive） k-alliance free, if for all defensive （offensive） k-alliance S, S／ X ≠ 0, i.e., X does not contain any defensive （offensive） k-alliance as a subset. A set Y C V is a defensive （offensive） k-alliance cover, if for all defensive （offensive） k-alliance S, S ∩ Y ≠ 0, i.e., Y contains at least one vertex from each defensive （offensive） k-alliance of F. 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.
作者 Juan Alberto RODRIGUEZ-VELAZQUEZ Jose Maria SIGARRETA Ismael GONZALEZ YERO Sergio BERMUDO
机构地区 Departament d'Enginyeria Informdtica i Matematiques Faculty of Mathematics Departament d'Enginyeria Informatica i Matematiques Department of Economy
出处 《Acta Mathematica Sinica,English Series》 SCIE CSCD 2011年第3期497-504,共8页 数学学报（英文版）
关键词 Defensive alliance offensive alliance alliance free set alliance cover set Defensive alliance, offensive alliance, alliance free set, alliance cover set
分类号 N031 [自然科学总论—科学技术哲学] G237.5 [文化科学]
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  • 1Kristiansen, P., Hedetniemi, S. M., Hedetniemi, S. T.: Alliances in graphs. Journal of Combinatorial Mathematics and Combinatorial Computing, 48, 157-177 (2004).
  • 2Brigham, R. C., Dutton, R. D., Hedetniemi, S. T.: A sharp lower bound on the powerful alliance number of Cm χ Cn. Congressus Numerantium, 167, 57-63 (2004).
  • 3Brigham, R. C., Dutton, R. D., Haynes, T. W., et al.: Powerful alliances in graphs. Discrete Mathematics, 309(8), 2140-2147 (2009).
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Acta Mathematica Sinica,English Series
2011年 第3期

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