Normal Helly circular-arc graphs and its subclasses

  • Authors:
  • Min Chih Lin;Francisco J. Soulignac;Jayme L. Szwarcfiter

  • Affiliations:
  • CONICET and Instituto de Cálculo, FCEN, Universidad de Buenos Aires, Buenos Aires, Argentina;CONICET and Departamento de Computación, FCEN, Universidad de Buenos Aires, Buenos Aires, Argentina;Universidade Federal do Rio de Janeiro, Instituto de Matemática, NCE and COPPE, Caixa Postal 2324, 20001-970 Rio de Janeiro, RJ, Brazil

  • Venue:
  • Discrete Applied Mathematics
  • Year:
  • 2013

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Abstract

A Helly circular-arc model M=(C,A) is a circle C together with a Helly family A of arcs of C. If no arc is contained in any other, then M is a proper Helly circular-arc model, if every arc has the same length, then M is a unit Helly circular-arc model, and if there are no two arcs covering the circle, then M is a normal Helly circular-arc model. A Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc graph is the intersection graph of the arcs of a Helly (resp. proper Helly, unit Helly, normal Helly) circular-arc model. In this article we study these subclasses of Helly circular-arc graphs. We show natural generalizations of several properties of (proper) interval graphs that hold for some of these Helly circular-arc subclasses. Next, we describe characterizations for the subclasses of Helly circular-arc graphs, including forbidden induced subgraphs characterizations. These characterizations lead to efficient algorithms for recognizing graphs within these classes. Finally, we show how these classes of graphs relate with straight and round digraphs.