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k-Domination and k-Independence in Graphs: A Survey

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Abstract

In 1985, Fink and Jacobson gave a generalization of the concepts of domination and independence in graphs. For a positive integer k, a subset S of vertices in a graph G = (V, E) is k-dominating if every vertex of V − S is adjacent to at least k vertices in S. The subset S is k-independent if the maximum degree of the subgraph induced by the vertices of S is less or equal to k − 1. In this paper we survey results on k-domination and k-independence.

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Authors and Affiliations

  1. LAMDA-RO Laboratory, Department of Mathematics, University of Blida, B.P. 270, Blida, Algeria

    Mustapha Chellali

  2. LRI, URM 8623, University Paris-Sud and CNRS, 91405, Orsay Cedex, France

    Odile Favaron

  3. Dep. Matemàtica Aplicada III, Edifici C2, UPC Barcelona, C/ Jordi Girona 1 i 3, 08034, Barcelona, Spain

    Adriana Hansberg

  4. Lehrstuhl II für Mathematik, RWTH Aachen University, Templergraben 55, 52056, Aachen, Germany

    Lutz Volkmann

Authors
  1. Mustapha Chellali
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  2. Odile Favaron
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  3. Adriana Hansberg
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  4. Lutz Volkmann
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Corresponding author

Correspondence to Mustapha Chellali.

Additional information

M. Chellali was supported by “Programmes Nationaux de Recherche: Code 8/u09/510”.

A. Hansberg was partially supported by the Ministry of Science and Innovation, Spain, and the European Regional Development Fund (ERDF) under project MTM2008-066200-C03-02/MTM.

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Chellali, M., Favaron, O., Hansberg, A. et al. k-Domination and k-Independence in Graphs: A Survey. Graphs and Combinatorics 28, 1–55 (2012). https://doi.org/10.1007/s00373-011-1040-3

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  • DOI: https://doi.org/10.1007/s00373-011-1040-3

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