Acta Mathematica Sinica, English Series

, Volume 27, Issue 3, pp 497–504 | Cite as

Alliance free and alliance cover sets

  • Juan Alberto Rodriguez-Velazquez
  • José María Sigarreta
  • Ismael Gonzalez Yero
  • Sergio Bermudo
Article

Abstract

A defensive (offensive) k-alliance in Γ = (V,E) is a set SV 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 XV 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 YV is a defensive (offensive) k-alliance cover, if for all defensive (offensive) k-alliance S, SY ≠ ∅, 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.

Keywords

Defensive alliance offensive alliance alliance free set alliance cover set 

MR(2000) Subject Classification

05C69 05C70 

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Copyright information

© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Juan Alberto Rodriguez-Velazquez
    • 1
  • José María Sigarreta
    • 2
  • Ismael Gonzalez Yero
    • 1
  • Sergio Bermudo
    • 3
  1. 1.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain
  2. 2.Faculty of MathematicsAutonomous University of GuerreroAcapulco, GuerreroMexico
  3. 3.Department of Economy, Quantitative Methods and Economic HistoryPablo de Olavide UniversitySevillaSpain

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