Triangle Contact Representations and Duality
A contact representation by triangles of a graph is a set of triangles in the plane such that two triangles intersect on at most one point, each triangle represents a vertex of the graph and two triangles intersects if and only if their corresponding vertices are adjacent. de Fraysseix, Ossona de Mendez and Rosenstiehl proved that every planar graph admits a contact representation by triangles. We strengthen this in terms of a simultaneous contact representation by triangles of a planar map and of its dual.
A primal-dual contact representation by triangles of a planar map is a contact representation by triangles of the primal and a contact representation by triangles of the dual such that for every edge uv, bordering faces f and g, the intersection between the triangles corresponding to u and v is the same point as the intersection between the triangles corresponding to f and g. We prove that every 3-connected planar map admits a primal-dual contact representation by triangles. Moreover, the interiors of the triangles form a tiling of the triangle corresponding to the outer face and each contact point is a node of exactly three triangles. Then we show that these representations are in one-to-one correspondence with generalized Schnyder woods defined by Felsner for 3-connected planar maps.
- 1.Andreev, E.: On convex polyhedra in Lobachevsky spaces. In: Mat. Sbornik, Ser., vol. 81, pp. 445–478 (1970)Google Scholar
- 2.Badent, M., Binucci, C., Di Giacomo, E., Didimo, W., Felsner, S., Giordano, F., KratochvÃI, J. Palladino, P., Patrignani, M., Trotta., F.: Homothetic triangle contact representations of planar graphs. In: Proceedings of the 19th Canadian Conference on Computational Geometry CCCG 2007, pp. 233–236 (2007)Google Scholar
- 6.Felsner, S.: Lattice structures from planar graphs. Electron. J. Combin. 11 (2004)Google Scholar
- 12.Gansner, E.R., Hu, Y., Kobourov, S.G.: On Touching Triangle Graphs. In: Proc. Graph Drawing (2010), http://arxiv1.library.cornell.edu/abs/1001.2862v1
- 13.Kaufmann, M., Kratochvíl, J., Lehmann, K.A., Subramanian, A.R.: Max-tolerance graphs as intersection graphs: Cliques, cycles and recognition. In: Proc. SODA 2006, pp. 832–841 (2006)Google Scholar
- 15.Kratochvíl, J.: Bertinoro Workshop on Graph Drawing (2007)Google Scholar
- 18.Schramm, O.: Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps, Modified version of PhD thesis from (1990), http://arxiv.org/abs/0709.0710v1