Extending Partial Representations of Proper and Unit Interval Graphs

  • Pavel Klavík
  • Jan Kratochvíl
  • Yota Otachi
  • Ignaz Rutter
  • Toshiki Saitoh
  • Maria Saumell
  • Tomáš Vyskočil
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)


The recently introduced problem of extending partial interval representations asks, for an interval graph with some intervals pre-drawn by the input, whether the partial representation can be extended to a representation of the entire graph. In this paper, we give a linear-time algorithm for extending proper interval representations and an almost quadratic-time algorithm for extending unit interval representations.

We also introduce the more general problem of bounded representations of unit interval graphs, where the input constrains the positions of intervals by lower and upper bounds. We show that this problem is Open image in new window -complete for disconnected input graphs and give a polynomial-time algorithm for a special class of instances, where the ordering of the connected components of the input graph along the real line is fixed. This includes the case of partial representation extension.

The hardness result sharply contrasts the recent polynomial-time algorithm for bounded representations of proper interval graphs [Balko et al. ISAAC’13]. So unless Open image in new window , proper and unit interval representations have very different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.


Intersection Graph Interval Graph Input Graph Full Version Chordal Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Pavel Klavík
    • 1
  • Jan Kratochvíl
    • 2
  • Yota Otachi
    • 3
  • Ignaz Rutter
    • 4
    • 2
  • Toshiki Saitoh
    • 5
  • Maria Saumell
    • 6
  • Tomáš Vyskočil
    • 2
  1. 1.Computer Science Institute, Faculty of Mathematics and PhysicsCharles University in PragueCzech Republic
  2. 2.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles University in PragueCzech Republic
  3. 3.School of Information ScienceJapan Advanced Institute of Science and TechnologyJapan
  4. 4.Institute of Theoretical Informatics, Faculty of InformaticsKarlsruhe Institute of Technology (KIT)Germany
  5. 5.Graduate School of EngineeringKobe UniversityKobeJapan
  6. 6.Department of MathematicsUniversity of West BohemiaPlzeňCzech Republic

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