, Volume 77, Issue 4, pp 1071–1104 | Cite as

Extending Partial Representations of Proper and Unit Interval Graphs

  • Pavel Klavík
  • Jan Kratochvíl
  • Yota Otachi
  • Ignaz Rutter
  • Toshiki Saitoh
  • Maria Saumell
  • Tomáš Vyskočil


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 some intervals by lower and upper bounds. We show that this problem is NP-complete for disconnected input graphs and give a polynomial-time algorithm for the special class of instances, where the ordering of the connected components of the input graph along the real line is prescribed. 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. in 2013). So unless \({\textsf {P}} = {\textsf {NP}}\), proper and unit interval representations have vastly different structure. This explains why partial representation extension problems for these different types of representations require substantially different techniques.


Intersection representation Partial representation extension Bounded representations Restricted representation Proper interval graph Unit interval graph  Linear programming 



We would like to thank an anonymous reviewer for suggestions which improved understandability of Sect. 2. The first, second and sixth authors are supported by ESF Eurogiga project GraDR as GAČR GIG/11/E023, the first two authors also by GAČR 14-14179S and Charles University as GAUK 196213. The fourth author is supported by a fellowship within the Postdoc-Program of the German Academic Exchange Service (DAAD). The sixth author is supported by the project LO1506 of the Czech Ministry of Education, Youth and Sports, the project NEXLIZ—CZ.1.07/2.3.00/30.0038, which is co-financed by the European Social Fund and the state budget of the Czech Republic, and ESF EuroGIGA project ComPoSe as F.R.S.-FNRS—EUROGIGA NR 13604.


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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
  • Ignaz Rutter
    • 2
    • 4
  • Toshiki Saitoh
    • 5
  • Maria Saumell
    • 6
  • Tomáš Vyskočil
    • 7
  1. 1.Computer Science InstituteCharles 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.Faculty of InformaticsKarlsruhe Institute of TechnologyKarlsruheGermany
  5. 5.Graduate School of EngineeringKobe UniversityNadaJapan
  6. 6.Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society)University of West BohemiaPlzeňCzech Republic
  7. 7.Department of Computer ScienceRutgers, The State University of New JerseyPiscatawayUSA

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