GD 2010: Graph Drawing pp 377-388

Convex Polygon Intersection Graphs

• Erik Jan van Leeuwen
• Jan van Leeuwen
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6502)

Abstract

Geometric intersection graphs are graphs determined by intersections of geometric objects. We study the complexity of visualizing the arrangements of objects that induce such graphs. We give a general framework for describing geometric intersection graphs, using arbitrary finite base sets of rationally given convex polygons and affine transformations. We prove that for every class of intersection graphs that fits the framework, the graphs in the class have a representation using polynomially many bits. Consequently, the recognition problem of these classes is in NP (and thus NP-complete). We also give an algorithm to find a drawing of the objects in the plane, if a graph class fits the framework.

References

1. 1.
H.: Breu, Algorithmic aspects of constrained unit disk graphs, PhD Thesis, The University of British Columbia, Vancouver (1996)Google Scholar
2. 2.
Breu, H., Kirkpatrick, D.G.: Unit disk graph recognition is NP-hard. Computational Geometry 9, 3–24 (1998)
3. 3.
Brightwell, G.R., Scheinerman, E.R.: Representations of planar graphs. SIAM Journal of Discrete Mathematics 6(2), 214–229 (1993)
4. 4.
Czyzowicz, J., Kranakis, E., Krizanc, D., Urrutia, J.: Discrete realizations of contact and intersection graphs. Int. J. Pure and Applied Mathematics 13(4), 429–442 (2004)
5. 5.
Deng, X., Hell, P., Huang, J.: Linear time representation of proper circular arc graphs and proper interval graphs. SIAM Journal of Computing 25, 390–403 (1996)
6. 6.
Edelsbrunner, H.: Computing the extreme distances between two convex polygons. J. of Algorithms 6, 213–224 (1985)
7. 7.
Golumbic, M.C., Trenk, A.N.: Tolerance graphs. Cambridge University Press, Cambridge (2004)
8. 8.
Hayward, R.B., Shamir, R.: A note on tolerance graph recognition. Discrete Applied Mathematics 143, 307–311 (2004)
9. 9.
Hliněný, P., Kratochvíl, J.: Representing graphs by disks and balls (A survey of recognition-complexity results). Discrete Mathematics 229, 101–124 (2001)
10. 10.
Kaufmann, M., Kratochvíl, J., Lehmann, K.A., Subramanian, A.R.: Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition. In: Proc. 17th Ann. ACM-SIAM Symp. on Discrete Algorithms (SODA 2006), pp. 832–841 (2006)Google Scholar
11. 11.
Kozyrev, V.P., Yushmanov, S.V.: Representations of graphs and networks (codings, layouts and embeddings). Journal of Soviet Mathematics 61(3), 2152–2194 (1992)
12. 12.
Kratochvíl, J., Matoušek, J.: NP-hardness results for intersection graphs. Commentationes Mathematicae Universitatis Carolinae 30(4), 761–773 (1989)
13. 13.
Kratochvíl, J.: A special planar satisfiability problem and a consequence of its NP-completeness. Discrete Applied Mathematics 52(3), 233–252 (1994)
14. 14.
Kratochvíl, J.: Intersection graphs of noncrossing arc-connected sets in the plane. In: North, S.C. (ed.) GD 1996. LNCS, vol. 1190, pp. 257–270. Springer, Heidelberg (1997)
15. 15.
Kratochvíl, J.: Geometric representations of graphs, Graduate Course, notes, Universitat Politècnica de Catalunya, Barcelona (April 2005), http://www.aco.gatech.edu/conference/archive/acokratochvil.ppt
16. 16.
Kratochvíl, J., Pergel, M.: Intersection graphs of homothetic polygons. In: Electronic Notes in Discrete Mathematics, vol. 31, pp. 277–280 (2008), http://www.canalc2.tv/video.asp?idvideo=7571
17. 17.
Lin, M.C., Szwarcfiter, J.L.: Unit circular-arc graph representations and feasible circulations. SIAM J. Discrete Mathematics 22(1), 409–423 (2008)
18. 18.
Lingas, A., Wahlen, M.: A note on maximum independent set and related problems on box graphs. Inf. Proc. Letters 93, 169–171 (2005)
19. 19.
McDiarmid, C., Müller, T.: The number of bits needed to represent a unit disk graph. In: Thilikos, D.M. (ed.) WG 2010. LNCS, vol. 6410, pp. 315–323. Springer, Heidelberg (2010)
20. 20.
McKee, T.A., Mc̨Morris, F.R.: Topics in intersection graph theory. SIAM Monographs on Discrete Mathematics and Applications, vol. 2, SIAM, Philadelphia (1999)Google Scholar
21. 21.
Pergel, M.: Special graph classes and algorithms on them, PhD Thesis, Dept. of Applied Mathematics, Charles University, Prague (2008)Google Scholar
22. 22.
Spinrad, J.R.: Efficient graph representations. In: Field Institute Monographs, vol. 19, American Mathematical Society, Providence (2003)Google Scholar
23. 23.
van Leeuwen, E.J.: Optimization and approximation on systems of geometric objects, PhD thesis, University of Amsterdam (2009)Google Scholar
24. 24.
van Leeuwen, E.J., van Leeuwen, J.: On the representation of disk graphs, Techn. Report UU-CS-2006-037, Dept. of Information and Computing Sciences, Utrecht University (2006)Google Scholar
25. 25.
van Leeuwen, E.J., van Leeuwen, J.: Convex polygon intersection graphs, Techn. Report, Dept. of Information and Computing Sciences, Utrecht University (to appear, 2010)Google Scholar