Acta Mathematica Sinica, English Series

, Volume 27, Issue 1, pp 73–82 | Cite as

Partitioning a graph into defensive k-alliances

  • Ismael G. Yero
  • Sergio Bermudo
  • Juan A. Rodríguez-Velázquez
  • José M. Sigarreta
Article

Abstract

A defensive k-alliance in a graph is a set S of vertices with the property that every vertex in S has at least k more neighbors in S than it has outside of S. A defensive k-alliance S is called global if it forms a dominating set. In this paper we study the problem of partitioning the vertex set of a graph into (global) defensive k-alliances. The (global) defensive k-alliance partition number of a graph Θ = (V, E), (ψkgd (Γ)) ψkd(Γ), is defined to be the maximum number of sets in a partition of V such that each set is a (global) defensive k-alliance. We obtain tight bounds on ψkd(Θ) and ψkgd (Γ) in terms of several parameters of the graph including the order, size, maximum and minimum degree, the algebraic connectivity and the isoperimetric number. Moreover, we study the close relationships that exist among partitions of Γ1 × Γ2 into (global) defensive (k1 + k2)-alliances and partitions of Γi into (global) defensive ki-alliances, i ∈ {1, 2}.

Keywords

Defensive alliances dominating sets domination isoperimetric number 

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

  • Ismael G. Yero
    • 1
  • Sergio Bermudo
    • 2
  • Juan A. Rodríguez-Velázquez
    • 3
  • José M. Sigarreta
    • 4
  1. 1.Department of Computer Engineering and MathematicsRovira i Virgili UniversityTarragonaSpain
  2. 2.Department of Economy, Quantitative Methods and Economic HistoryPablo de Olavide UniversitySevillaSpain
  3. 3.Department of Computer Engineering and MathematicsRovira i Virgili UniversityTarragonaSpain
  4. 4.Faculty of MathematicsAutonomous University of GuerreroAcapulco, GuerreroMéxico

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