, Volume 59, Issue 2, pp 215–239 | Cite as

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

  • Benson L. Joeris
  • Min Chih LinEmail author
  • Ross M. McConnell
  • Jeremy P. Spinrad
  • Jayme L. Szwarcfiter


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.


Algorithms Circular-arc graphs Forbidden subgraphs Helly circular-arc graphs 


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Copyright information

© Springer Science+Business Media, LLC 2009

Authors and Affiliations

  • Benson L. Joeris
    • 1
  • Min Chih Lin
    • 2
    Email author
  • Ross M. McConnell
    • 1
  • Jeremy P. Spinrad
    • 3
  • Jayme L. Szwarcfiter
    • 4
  1. 1.Computer Science DepartmentColorado State UniversityFort CollinsUSA
  2. 2.Facultad de Ciencias Exactas y Naturales, Departamento de ComputaciónUniversidad de Buenos AiresBuenos AiresArgentina
  3. 3.Computer Science DepartmentVanderbilt UniversityNashvilleUSA
  4. 4.Instituto de Matemática, NCE and COPPEUniversidade Federal do Rio de JaneiroRio de JaneiroBrazil

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