ESA 2012: Algorithms – ESA 2012 pp 671-682

# Extending Partial Representations of Function Graphs and Permutation Graphs

• Pavel Klavík
• Jan Kratochvíl
• Tomasz Krawczyk
• Bartosz Walczak
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7501)

## Abstract

Function graphs are graphs representable by intersections of continuous real-valued functions on the interval [0,1] and are known to be exactly the complements of comparability graphs. As such they are recognizable in polynomial time. Function graphs generalize permutation graphs, which arise when all functions considered are linear.

We focus on the problem of extending partial representations, which generalizes the recognition problem. We observe that for permutation graphs an easy extension of Golumbic’s comparability graph recognition algorithm can be exploited. This approach fails for function graphs. Nevertheless, we present a polynomial-time algorithm for extending a partial representation of a graph by functions defined on the entire interval [0,1] provided for some of the vertices. On the other hand, we show that if a partial representation consists of functions defined on subintervals of [0,1], then the problem of extending this representation to functions on the entire interval [0,1] becomes NP-complete.

## Preview

Unable to display preview. Download preview PDF.

### References

1. 1.
Angelini, P., Di Battista, G., Frati, F., Jelínek, V., Kratochvíl, J., Patrignani, M., Rutter, I.: Testing planarity of partially embedded graphs. In: SODA 2010, pp. 202–221 (2010)Google Scholar
2. 2.
Aspvall, B., Plass, M.F., Tarjan, R.E.: A linear-time algorithm for testing the truth of certain quantified boolean formulas. Inform. Process. Lett. 8, 121–123 (1979)
3. 3.
Booth, K.S., Lueker, G.S.: Testing for the consecutive ones property, interval graphs, and planarity using PQ-tree algorithms. J. Comput. Sys. Sci. 13, 335–379 (1976)
4. 4.
Bläsius, T., Rutter, I.: Simultaneous PQ-ordering with applications to constrained embedding problems (2011), http://arxiv.org/abs/1112.0245
5. 5.
Even, S., Pnueli, A., Lempel, A.: Permutation graphs and transitive graphs. J. ACM 19, 400–410 (1972)
6. 6.
Fiala, J.: NP-completeness of the edge precoloring extension problem on bipartite graphs. J. Graph Theory 43, 156–160 (2003)
7. 7.
Gallai, T.: Transitiv orientierbare Graphen. Acta Mathematica Academiae Scientiarum Hungaricae 18, 25–66 (1967)
8. 8.
Golumbic, M.C.: The complexity of comparability graph recognition and coloring. Computing 18, 199–208 (1977)
9. 9.
Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press (1980)Google Scholar
10. 10.
Golumbic, M.C., Rotem, D., Urrutia, J.: Comparability graphs and intersection graphs. Discrete Math. 43, 37–46 (1983)
11. 11.
Jampani, K.R., Lubiw, A.: Simultaneous Interval Graphs. In: Cheong, O., Chwa, K.-Y., Park, K. (eds.) ISAAC 2010. LNCS, vol. 6506, pp. 206–217. Springer, Heidelberg (2010)
12. 12.
Jampani, K.R., Lubiw, A.: The simultaneous representation problem for chordal, comparability and permutation graphs. Graph Algorithms Appl. 16, 283–315 (2012)
13. 13.
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)
14. 14.
Kratochvíl, J.: String graphs. II. recognizing string graphs is NP-hard. J. Combin. Theory Ser. B 52, 67–78 (1991)
15. 15.
Krom, M.R.: The decision problem for a class of first-order formulas in which all disjunctions are binary. Zeitschrift für Mathematische Logik und Grundlagen der Mathematik 13, 15–20 (1967)
16. 16.
Kuratowski, K.: Sur le problème des courbes gauches en topologie. Fund. Math. 15, 217–283 (1930)Google Scholar
17. 17.
Marx, D.: NP-completeness of list coloring and precoloring extension on the edges of planar graphs. J. Graph Theory 49, 313–324 (2005)
18. 18.
McConnell, R.M., Spinrad, J.P.: Modular decomposition and transitive orientation. Discrete Math. 201, 189–241 (1999)
19. 19.
Patrignani, M.: On Extending a Partial Straight-Line Drawing. In: Healy, P., Nikolov, N.S. (eds.) GD 2005. LNCS, vol. 3843, pp. 380–385. Springer, Heidelberg (2006)
20. 20.
Schaefer, M., Sedgwick, E., Štefankovič, D.: Recognizing string graphs in NP. In: STOC 2002, pp. 1–6 (2002)Google Scholar
21. 21.
Spinrad, J.P.: Efficient Graph Representations. Field Institute Monographs (2003)Google Scholar

## Copyright information

© Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

• Pavel Klavík
• 1
• Jan Kratochvíl
• 1
• Tomasz Krawczyk
• 2
• Bartosz Walczak
• 2
1. 1.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityCzech Republic
2. 2.Theoretical Computer Science Department, Faculty of Mathematics and Computer ScienceJagiellonian UniversityPoland