k-Gap Interval Graphs

  • Fedor V. Fomin
  • Serge Gaspers
  • Petr Golovach
  • Karol Suchan
  • Stefan Szeider
  • Erik Jan van Leeuwen
  • Martin Vatshelle
  • Yngve Villanger
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7256)

Abstract

We initiate the study of a new parameterization of graph problems. In a multiple interval representation of a graph, each vertex is associated to at least one interval of the real line, with an edge between two vertices if and only if an interval associated to one vertex has a nonempty intersection with an interval associated to the other vertex. A graph on n vertices is a k-gap interval graph if it has a multiple interval representation with at most n + k intervals in total. In order to scale up the nice algorithmic properties of interval graphs (where k = 0), we parameterize graph problems by k, and find FPT algorithms for several problems, including Feedback Vertex Set, Dominating Set, Independent Set, Clique, Clique Cover, and Multiple Interval Transversal. The Coloring problem turns out to be Open image in new window-hard and we design an XP algorithm for the recognition problem.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Fedor V. Fomin
    • 1
  • Serge Gaspers
    • 2
  • Petr Golovach
    • 3
  • Karol Suchan
    • 4
    • 5
  • Stefan Szeider
    • 2
  • Erik Jan van Leeuwen
    • 1
  • Martin Vatshelle
    • 1
  • Yngve Villanger
    • 1
  1. 1.Department of InformaticsUniversity of BergenBergenNorway
  2. 2.Inst. of Information SystemsVienna University of TechnologyViennaAustria
  3. 3.School of Engineering and Computing SciencesDurham UniversityDurhamUK
  4. 4.Facultad de Ingeniería y CienciasUniversidad Adolfo IbáñezSantiagoChile
  5. 5.Faculty of Applied Mathematics WMSAGH - University of Science and TechnologyKrakowPoland

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