Abstract
A touching triangle graph (TTG) representation of a planar graph is a planar drawing Γ of the graph, where each vertex is represented as a triangle and each edge e is represented as a side contact of the triangles that correspond to the end vertices of e. We call Γ a proper TTG representation if Γ determines a tiling of a triangle, where each tile corresponds to a distinct vertex of the input graph. In this paper we prove that every 3-connected cubic planar graph admits a proper TTG representation. We also construct proper TTG representations for parabolic grid graphs and the graphs determined by rectangular grid drawings (e.g., square grid graphs). Finally, we describe a fixed-parameter tractable decision algorithm for testing whether a 3-connected planar graph admits a proper TTG representation.
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
Batagelj, V.: Inductive classes of cubic graphs. In: Proceedings of the 6th Hungarian Colloquium on Combinatorics, Eger, Hungary. Finite and infinite sets, vol. 37, pp. 89–101 (1981)
Buchsbaum, A., Gansner, E., Procopiuc, C., Venkatasubramanian, S.: Rectangular layouts and contact graphs. ACM Transactions on Algorithms 4(1) (2008)
de Fraysseix, H., de Mendez, P.O.: Barycentric systems and stretchability. Discrete Applied Mathematics 155(9), 1079–1095 (2007)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10, 41–51 (1990)
de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Combinatorics, Probability & Computing 3, 233–246 (1994)
Duncan, C., Gansner, E.R., Hu, Y., Kaufmann, M., Kobourov, S.G.: Optimal polygonal representation of planar graphs. Algorithmica 63(3), 672–691 (2012)
Gansner, E.R., Hu, Y., Kobourov, S.G.: On Touching Triangle Graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 250–261. Springer, Heidelberg (2011)
He, X.: On floor-plan of plane graphs. SIAM Journal on Computing 28(6), 2150–2167 (1999)
Koebe, P.: Kontaktprobleme der konformen Abbildung. Berichte über die Verhandlungen der Sächsischen Akademie der Wissenschaften zu Leipzig. Math.-Phys. Klasse 88, 141–164 (1936)
Liao, C.C., Lu, H.I., Yen, H.C.: Compact floor-planning via orderly spanning trees. Journal of Algorithms 48, 441–451 (2003)
Phillips, R.: The Order-5 triangle partitions, http://www.mathpuzzle.com/triangle.html (accessed June 7, 2012)
Rahman, M., Nishizeki, T., Ghosh, S.: Rectangular drawings of planar graphs. Journal of Algorithms 50(1), 62–78 (2004)
Thomassen, C.: Interval representations of planar graphs. Journal of Combinatorial Theory (B) 40(1), 9–20 (1986)
Woeginger, G.J.: Exact Algorithms for NP-Hard Problems: A Survey. In: Jünger, M., Reinelt, G., Rinaldi, G. (eds.) Combinatorial Optimization - Eureka, You Shrink! LNCS, vol. 2570, pp. 185–207. Springer, Heidelberg (2003)
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
Kobourov, S.G., Mondal, D., Nishat, R.I. (2013). Touching Triangle Representations for 3-Connected Planar 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_18
Download citation
DOI: https://doi.org/10.1007/978-3-642-36763-2_18
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)