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Alliance free and alliance cover sets

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Abstract

A defensive (offensive) k-alliance in Γ = (V,E) is a set S ⊆ V 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 X ⊆ V 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 Y ⊆ V is a defensive (offensive) k-alliance cover, if for all defensive (offensive) k-alliance S, S ∩ Y ≠ ∅, 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.

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References

  1. Hedetniemi, S. M., Hedetniemi, S. T., Kristiansen, P.: Alliances in graphs. J. Combin. Math. Combin. Comput., 48, 157–177 (2004)

    MATH  MathSciNet  Google Scholar 

  2. Shafique, K. H., Dutton, R. D.: Maximum alliance-free and minimum alliance-cover sets. Congr. Numer., 162, 139–146 (2003)

    MATH  MathSciNet  Google Scholar 

  3. Fernau, H., Rodríguez, J. A., Sigarreta, J. M.: Offensive k-alliances in graphs. Discrete Appl. Math., 157(2), 177–182 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  4. Rodríguez, J. A., Yero, I. G., Sigarreta, J. M.: Defensive k-alliances in graphs. Appl. Math. Lett., 22, 96–100 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Rodríguez-Velázquez, J. A., Sigarreta, J. M.: Global defensive k-alliances in graphs. Discrete Appl. Math., 157(2), 211–218 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  6. Sigarreta, J. M., Rodríguez, J. A.: On the global offensive alliance number of a graph. Discrete Appl. Math., 157(2), 219–226 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  7. Sigarreta, J. M., Rodríguez, J. A.: On defensive alliances and line graphs. Appl. Math. Lett., 19(12), 1345–1350 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  8. Yero, I. G., Rodríguez-Velázquez, J. A.: Boundary defensive k-alliances in graphs. Discrete Appl. Math., 158, 1205–1211 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  9. Yero, I. G., Bermudo, S., Rodríguez-Velázquez, J. A., et al.: Partitioning a graph into defensive k-alliances. Acta Mathematica Sinica, English Series, 27(1), 73–82 (2011)

    Article  MathSciNet  Google Scholar 

  10. Shafique, K. H., Dutton, R.: A tight bound on the cardinalities of maximun alliance-free and minimun alliance-cover sets. J. Combin. Math. Combin. Comput., 56, 139–145 (2006)

    MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Departament d’Enginyeria Informàtica i Matemàtiques, Universitat Rovira i Virgili, Av. Països Catalans 26, 43007, Tarragona, Spain

    Juan Alberto Rodriguez-Velazquez & Ismael Gonzalez Yero

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

    José María Sigarreta

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

    Sergio Bermudo

Authors
  1. Juan Alberto Rodriguez-Velazquez
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  2. José María Sigarreta
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  3. Ismael Gonzalez Yero
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  4. Sergio Bermudo
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Corresponding author

Correspondence to Juan Alberto Rodriguez-Velazquez.

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, and by the Junta de Andalucía, ref. FQM-260 and ref. P06-FQM-02225

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Rodriguez-Velazquez, J.A., Sigarreta, J.M., Gonzalez Yero, I. et al. Alliance free and alliance cover sets. Acta. Math. Sin.-English Ser. 27, 497–504 (2011). https://doi.org/10.1007/s10114-011-0056-1

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  • DOI: https://doi.org/10.1007/s10114-011-0056-1

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