, Volume 78, Issue 3, pp 945–967 | Cite as

Extending Partial Representations of Interval Graphs

  • Pavel Klavík
  • Jan Kratochvíl
  • Yota Otachi
  • Toshiki Saitoh
  • Tomáš Vyskočil


Interval graphs are intersection graphs of closed intervals of the real-line. The well-known computational problem, called recognition, asks whether an input graph G can be represented by closed intervals, i.e., whether G is an interval graph. There are several linear-time algorithms known for recognizing interval graphs, the oldest one is by Booth and Lueker (J Comput Syst Sci 13:335–379, 1976) based on PQ-trees. In this paper, we study a generalization of recognition, called partial representation extension. The input of this problem consists of a graph G with a partial representation \({{{\mathcal {R}}}}'\) fixing the positions of some intervals. The problem asks whether it is possible to place the remaining interval and create an interval representation \({{{\mathcal {R}}}}\) of the entire graph G extending \({{{\mathcal {R}}}}'\). We generalize the characterization of interval graphs by Fulkerson and Gross (Pac J Math 15:835–855, 1965) to extendible partial representations. Using it, we give a linear-time algorithm for partial representation extension based on a reordering problem of PQ-trees.


Interval graphs Partial representation extension PQ-trees Linear-time algorithm 



We are very thankful to Pavol Hell for suggesting the PQ-trees approach, and to Martin Balko and Jiří Fiala for comments concerning writing. The first two authors are supported by CE-ITI (P202/12/G061 of GAČR) and Charles University as GAUK 196213.


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Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  • Pavel Klavík
    • 1
  • Jan Kratochvíl
    • 2
  • Yota Otachi
    • 3
  • Toshiki Saitoh
    • 4
  • Tomáš Vyskočil
    • 5
  1. 1.Computer Science Institute, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
  2. 2.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles University in PraguePragueCzech Republic
  3. 3.School of Information ScienceJapan Advanced Institute of Science and TechnologyNomiJapan
  4. 4.Graduate School of EngineeringKobe UniversityKobeJapan
  5. 5.Department of Computer ScienceRutgers, The State University of New JerseyPiscatawayUSA

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