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Graphs and Combinatorics

, Volume 28, Issue 4, pp 575–583 | Cite as

Partitioning a Graph into Global Powerful k-Alliances

  • Ismael G. Yero
  • Juan A. Rodríguez-Velázquez
Original Paper

Abstract

A set S of vertices of a graph is a defensive k-alliance if every vertex \({v\in S}\) has at least k more neighbors in S than it has outside of S. Analogously, a set S is an offensive k-alliance if every vertex in the neighborhood of S has at least k more neighbors in S than it has outside of S. Also, a powerful k-alliance is a set S of vertices of the graph, which is both defensive k-alliance and offensive (k + 2)-alliance. A powerful k-alliance is called global if it is a dominating set. In this paper we show that for k ≥ 0, no graph is partitionable into global powerful k-alliances and, for k ≤ −1, we obtain upper bounds on the maximum number of sets belonging to a partition of a graph into global powerful k-alliances. In addition, we study the close relationships that exist between partitions of a Cartesian product graph, Γ1 × Γ2, into (global) powerful (k 1 + k 2)-alliances and partitions of Γ i into (global) powerful k i -alliances, \({i\in \{1,2\}}\).

Keywords

Defensive k-alliances Offensive k-alliances Powerful k-alliances Cartesian product graphs 

Mathematics Subject Classification (2000)

05C69 05C70 

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Copyright information

© Springer 2011

Authors and Affiliations

  • Ismael G. Yero
    • 1
  • Juan A. Rodríguez-Velázquez
    • 1
  1. 1.Departament d’Enginyeria Informàtica i MatemàtiquesUniversitat Rovira i VirgiliTarragonaSpain

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