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Graphs with the Strong Havel–Hakimi Property

Graphs and Combinatorics volume 32, pages 1689–1697 (2016)Cite this article

Abstract

The Havel–Hakimi algorithm iteratively reduces the degree sequence of a graph to a list of zeroes. As shown by Favaron, Mahéo, and Saclé, the number of zeroes produced, known as the residue, is a lower bound on the independence number of the graph. We say that a graph has the strong Havel–Hakimi property if in each of its induced subgraphs, deleting any vertex of maximum degree reduces the degree sequence in the same way that the Havel–Hakimi algorithm does. We characterize graphs having this property (which include all threshold and matrogenic graphs) in terms of minimal forbidden induced subgraphs. We further show that for these graphs the residue equals the independence number, and a natural greedy algorithm always produces a maximum independent set.

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Acknowledgments

The authors wish to thank the anonymous referees for many useful comments that have improved the paper.

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Authors and Affiliations

  1. Department of Mathematics, University of Rhode Island, Kingston, RI, 02881, USA

    Michael D. Barrus

  2. Department of Mathematics, Brigham Young University, Provo, UT, 84602, USA

    Grant Molnar

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  1. Michael D. Barrus
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  2. Grant Molnar
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Correspondence to Michael D. Barrus.

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Barrus, M.D., Molnar, G. Graphs with the Strong Havel–Hakimi Property. Graphs and Combinatorics 32, 1689–1697 (2016). https://doi.org/10.1007/s00373-015-1674-7

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  • DOI: https://doi.org/10.1007/s00373-015-1674-7

Keywords

  • Independence number
  • Havel–Hakimi algorithm
  • Residue

Mathematics Subject Classification

  • 05C69
  • 05C07