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Extending Partial Representations of Interval Graphs

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Abstract

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.

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Notes

  1. For the purpose of Sect. 3, we allow empty intervals with \(\ell _v = r_v\).

  2. This can be done in constant time if we remember in each moment the positions of the two leftmost lower handles in the ordering, and update this information after removing one of them from .

  3. We also need to update here since it might happen that the interval is empty for some maximal clique a. This can happen only if some pre-drawn interval of P(a) is a singleton.

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Acknowledgments

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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Authors and Affiliations

  1. Computer Science Institute, Faculty of Mathematics and Physics, Charles University in Prague, Malostranské náměstí 25, 118 00, Prague, Czech Republic

    Pavel Klavík

  2. Department of Applied Mathematics, Faculty of Mathematics and Physics, Charles University in Prague, Malostranské náměstí 25, 118 00, Prague, Czech Republic

    Jan Kratochvíl

  3. School of Information Science, Japan Advanced Institute of Science and Technology, Asahidai 1-1, Nomi, Ishikawa, 923-1292, Japan

    Yota Otachi

  4. Graduate School of Engineering, Kobe University, Rokkodai 1-1, Nada, Kobe, 657-8501, Japan

    Toshiki Saitoh

  5. Department of Computer Science, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ, 08854-8019, USA

    Tomáš Vyskočil

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  1. Pavel Klavík
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Correspondence to Pavel Klavík.

Additional information

The conference version of this paper appeared in TAMC 2011 [22].

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Klavík, P., Kratochvíl, J., Otachi, Y. et al. Extending Partial Representations of Interval Graphs. Algorithmica 78, 945–967 (2017). https://doi.org/10.1007/s00453-016-0186-z

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