Abstract
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 -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 , 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.
For the full version of this paper, see arXiv:1207.6960.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: SODA 2010: Proc. 21st Annu. ACM-SIAM Sympos. Discr. Alg., pp. 202–221 (2010)
Balko, M., Klavík, P., Otachi, Y.: Bounded representations of interval and proper interval graphs. In: Cai, L., Cheng, S.-W., Lam, T.-W. (eds.) Algorithms and Computation. LNCS, vol. 8283, pp. 535–546. Springer, Heidelberg (2013)
Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems. In: SODA 2013: Proc. 24th Annu. ACM-SIAM Sympos. Discr. Alg., pp. 1030–1043 (2013)
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. System Sci. 13, 335–379 (1976)
Chaplick, S., Dorbec, P., Kratochvíl, J., Montassier, M., Stacho, J.: Contact representations of planar graph: Rebuilding is hard. In: WG 2014 (to appear, 2014)
Chaplick, S., Fulek, R., Klavík, P.: Extending partial representations of circle graphs. In: Wismath, S., Wolff, A. (eds.) GD 2013. LNCS, vol. 8242, pp. 131–142. Springer, Heidelberg (2013)
Cormen, T.H., Leiserson, C.E., Rivest, R.L., Stein, C.: Introduction to Algorithms, 3rd edn. The MIT Press (2009)
Corneil, D.G., Kim, H., Natarajan, S., Olariu, S., Sprague, A.P.: Simple linear time recognition of unit interval graphs. Inform. Process. Lett. 55(2), 99–104 (1995)
Corneil, D.G., Olariu, S., Stewart, L.: The LBFS structure and recognition of interval graphs. SIAM J. Discrete Math. 23(4), 1905–1953 (2009)
Deng, X., Hell, P., Huang, J.: Linear-time representation algorithms for proper circular-arc graphs and proper interval graphs. SIAM J. Comput. 25(2), 390–403 (1996)
Fürer, M.: Faster integer multiplication. SIAM J. Comput. 39(3), 979–1005 (2009)
Gilmore, P.C., Hoffman, A.J.: A characterization of comparability graphs and of interval graphs. Can. J. Math. 16, 539–548 (1964)
Hajós, G.: Über eine Art von Graphen. Internationale Mathematische Nachrichten 11, 65 (1957)
Jampani, K., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. J. Graph Algorithms Appl. 16(2), 283–315 (2012)
Klavík, P., Kratochvíl, J., Krawczyk, T., Walczak, B.: Extending partial representations of function graphs and permutation graphs. In: Epstein, L., Ferragina, P. (eds.) ESA 2012. LNCS, vol. 7501, pp. 671–682. Springer, Heidelberg (2012)
Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T.: Extending partial representations of subclasses of chordal graphs. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 444–454. Springer, Heidelberg (2012)
Klavík, P., Kratochvíl, J., Otachi, Y., Saitoh, T., Vyskočil, T.: Linear-time algorithm for partial representation extension of interval graphs (2012) (in preparation)
Klavík, P., Kratochvíl, J., Vyskočil, T.: Extending partial representations of interval graphs. In: Ogihara, M., Tarui, J. (eds.) TAMC 2011. LNCS, vol. 6648, pp. 276–285. Springer, Heidelberg (2011)
Patrignani, M.: On extending a partial straight-line drawing. Int. J. Found. Comput. Sci. 17(5), 1061–1070 (2006)
Roberts, F.S.: Indifference graphs. In: Harary, F. (ed.) Proof Techniques in Graph Theory, pp. 139–146. Academic Press (1969)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Klavík, P. et al. (2014). Extending Partial Representations of Proper and Unit Interval Graphs. In: Ravi, R., Gørtz, I.L. (eds) Algorithm Theory – SWAT 2014. SWAT 2014. Lecture Notes in Computer Science, vol 8503. Springer, Cham. https://doi.org/10.1007/978-3-319-08404-6_22
Download citation
DOI: https://doi.org/10.1007/978-3-319-08404-6_22
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-08403-9
Online ISBN: 978-3-319-08404-6
eBook Packages: Computer ScienceComputer Science (R0)