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Linear-Time Recognition of Helly Circular-Arc Models and Graphs

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Abstract

A circular-arc model ℳ is a circle C together with a collection \(\mathcal{A}\) of arcs of C. If \(\mathcal{A}\) satisfies the Helly Property then ℳ is a Helly circular-arc model. A (Helly) circular-arc graph is the intersection graph of a (Helly) circular-arc model. Circular-arc graphs and their subclasses have been the object of a great deal of attention in the literature. Linear-time recognition algorithms have been described both for the general class and for some of its subclasses. However, for Helly circular-arc graphs, the best recognition algorithm is that by Gavril, whose complexity is O(n 3). In this article, we describe different characterizations for Helly circular-arc graphs, including a characterization by forbidden induced subgraphs for the class. The characterizations lead to a linear-time recognition algorithm for recognizing graphs of this class. The algorithm also produces certificates for a negative answer, by exhibiting a forbidden subgraph of it, within this same bound.

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Authors and Affiliations

  1. Computer Science Department, Colorado State University, Fort Collins, CO, 80523-1873, USA

    Benson L. Joeris & Ross M. McConnell

  2. Facultad de Ciencias Exactas y Naturales, Departamento de Computación, Universidad de Buenos Aires, Buenos Aires, Argentina

    Min Chih Lin

  3. Computer Science Department, Vanderbilt University, Nashville, TN, 37235, USA

    Jeremy P. Spinrad

  4. Instituto de Matemática, NCE and COPPE, Universidade Federal do Rio de Janeiro, Caixa Postal 2324, 20001-970, Rio de Janeiro, RJ, Brazil

    Jayme L. Szwarcfiter

Authors
  1. Benson L. Joeris
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  2. Min Chih Lin
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  3. Ross M. McConnell
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  4. Jeremy P. Spinrad
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  5. Jayme L. Szwarcfiter
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Corresponding author

Correspondence to Min Chih Lin.

Additional information

M.C. Lin was partially supported by UBACyT Grants X456 and X143 and by PICT ANPCyT Grant 1562.

J.L. Szwarcfiter was partially supported by the Conselho Nacional de Desenvolvimento Científico e Tecnológico, CNPq, and Fundação de Amparo à Pesquisa do Estado do Rio de Janeiro, FAPERJ, Brasil.

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Joeris, B.L., Lin, M.C., McConnell, R.M. et al. Linear-Time Recognition of Helly Circular-Arc Models and Graphs. Algorithmica 59, 215–239 (2011). https://doi.org/10.1007/s00453-009-9304-5

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  • DOI: https://doi.org/10.1007/s00453-009-9304-5

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