Abstract
A graph is B k -VPG when it has an intersection representation by paths in a rectangular grid with at most k bends (turns). It is known that all planar graphs are B 3-VPG and this was conjectured to be tight. We disprove this conjecture by showing that all planar graphs are B 2-VPG. We also show that the 4-connected planar graphs are a subclass of the intersection graphs of Z-shapes (i.e., a special case of B 2-VPG). Additionally, we demonstrate that a B 2-VPG representation of a planar graph can be constructed in O(n 3/2) time. We further show that the triangle-free planar graphs are contact graphs of: L-shapes, Γ-shapes, vertical segments, and horizontal segments (i.e., a special case of contact B 1-VPG). From this proof we gain a new proof that bipartite planar graphs are a subclass of 2-DIR.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Alam, M.J., Biedl, T., Felsner, S., Kaufmann, M., Kobourov, S.G.: Proportional Contact Representations of Planar Graphs. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 26–38. Springer, Heidelberg (2012)
Asinowski, A., Cohen, E., Golumbic, M.C., Limouzy, V., Lipshteyn, M., Stern, M.: Vertex intersection graphs of paths on a grid. J. Graph Algorithms Appl. 16(2), 129–150 (2012)
Ben-Arroyo Hartman, I., Newman, I., Ziv, R.: On grid intersection graphs. Discrete Math. 87(1), 41–52 (1991)
de Castro, N., Cobos, F.J., Dana, J.C., Márquez, A., Noy, M.: Triangle-free planar graphs and segment intersection graphs. J. Graph Algorithms Appl. 6(1), 7–26 (2002)
Chalopin, J., Gonçalves, D., Ochem, P.: Planar graphs are in 1-STRING. In: 18th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 609–617 (2007)
Chalopin, J., Gonçalves, D.: Every planar graph is the intersection graph of segments in the plane: extended abstract. In: 41st Annual ACM Symposium on Theory of Computing, STOC 2009, pp. 631–638 (2009)
Ehrlich, G., Even, S., Tarjan, R.: Intersection graphs of curves in the plane. J. Comb. Theory, Ser. B 21(1), 8–20 (1976)
Eppstein, D.: Regular labelings and geometric structures. In: 22nd Canadian Conference on Computational Geometry, CCCG 2010, pp. 125–130 (2010)
de Fraysseix, H., Ossona de Mendez, P.: Representations by contact and intersection of segments. Algorithmica 47(4), 453–463 (2007)
de Fraysseix, H., Ossona de Mendez, P., Pach, J.: Representation of planar graphs by segments. Intuitive Geometry 63, 109–117 (1991)
de Fraysseix, H., Ossona de Mendez, P., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability & Computing 3, 233–246 (1994)
Fusy, E.: Combinatoire des cartes planaires et applications algorithmiques. PhD Thesis (2007)
Fusy, E.: Transversal structures on triangulations: A combinatorial study and straight-line drawings. Discrete Math. 309(7), 1870–1894 (2009)
Gavenčiak, T.: Private communication: Small planar graphs are L-graphs (2012)
He, X.: On finding the rectangular duals of planar triangular graphs. SIAM J. Comput. 22, 1218–1226 (1993)
Heldt, D., Knauer, K., Ueckerdt, T.: On the Bend-Number of Planar and Outerplanar Graphs. In: Fernández-Baca, D. (ed.) LATIN 2012. LNCS, vol. 7256, pp. 458–469. Springer, Heidelberg (2012)
Kant, G., He, X.: Regular edge labeling of 4-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci. 172(1-2), 175–193 (1997)
Kobourov, S., Ueckerdt, T., Verbeek, K.: Combinatorial and geometric properties of laman graphs (2012)
Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Math.-Phys. Klasse 88, 141–164 (1936)
Koźmiński, K., Kinnen, E.: Rectangular dual of planar graphs. Networks 15(2), 145–157 (1985)
Kratochvíl, J., Matoušek, J.: Intersection graphs of segments. J. Comb. Theory, Ser. B 62(2), 289–315 (1994)
Scheinerman, E.R.: Intersection Classes and Multiple Intersection Parameters of Graphs. Ph.D. thesis. Princeton University (1984)
Sinden, F.: Topology of thin film circuits. Bell System Tech. J. 45, 1639–1662 (1966)
Sun, Y., Sarrafzadeh, M.: Floorplanning by graph dualization: L-shaped modules. Algorithmica 10, 429–456 (1993)
Thomassen, C.: Interval representations of planar graphs. J. Comb. Theory, Ser. B 40(1), 9–20 (1986)
Ungar, P.: On diagrams representing graphs. J. London Math. Soc. 28, 336–342 (1953)
West, D.: Open problems. SIAM J. Discrete Math. Newslett. 2, 10–12 (1991)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2013 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Chaplick, S., Ueckerdt, T. (2013). Planar Graphs as VPG-Graphs. In: Didimo, W., Patrignani, M. (eds) Graph Drawing. GD 2012. Lecture Notes in Computer Science, vol 7704. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36763-2_16
Download citation
DOI: https://doi.org/10.1007/978-3-642-36763-2_16
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-36762-5
Online ISBN: 978-3-642-36763-2
eBook Packages: Computer ScienceComputer Science (R0)