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Partitioning a graph into defensive k-alliances

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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), (ψ gd k (Γ)) ψ d k (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 ψ d k (Θ) and ψ gd k (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 (k 1 + k 2)-alliances and partitions of Γ i into (global) defensive k i -alliances, i ∈ {1, 2}.

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Authors and Affiliations

  1. Department of Computer Engineering and Mathematics, Rovira i Virgili University, Av. Països Catalans 26, 43007, Tarragona, Spain

    Ismael G. Yero

  2. Department of Economy, Quantitative Methods and Economic History, Pablo de Olavide University, Carretera de Utrera Km. 1, 41013, Sevilla, Spain

    Sergio Bermudo

  3. Department of Computer Engineering and Mathematics, Rovira i Virgili University, Av. Països Catalans 26, 43007, Tarragona, Spain

    Juan A. Rodríguez-Velázquez

  4. Faculty of Mathematics, Autonomous University of Guerrero, Carlos E. Adame 5, Col. La Garita, Acapulco, Guerrero, México

    José M. Sigarreta

Authors
  1. Ismael G. Yero
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  2. Sergio Bermudo
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  3. Juan A. Rodríguez-Velázquez
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  4. José M. Sigarreta
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Correspondence to Ismael G. Yero.

Additional information

Supported by the Spanish Ministry of Science and Innovation through Projects TSI2007-65406-C03-01 “EAEGIS”, CONSOLIDER INGENIO 2010 CSD2007-0004 “ARES” and MTM2009-09501, by the Rovira i Virgili University through Project 2006AIRE-09 and by the Junta de Andalucía, ref. FQM-260 and ref. P06-FQM-02225

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Yero, I.G., Bermudo, S., Rodríguez-Velázquez, J.A. et al. Partitioning a graph into defensive k-alliances. Acta. Math. Sin.-English Ser. 27, 73–82 (2011). https://doi.org/10.1007/s10114-011-9075-1

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  • DOI: https://doi.org/10.1007/s10114-011-9075-1

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