Triangle Contact Representations and DualityGoncalves, Daniel and Leveque, Benjamin and Pinlou, Alexandre (2011) Triangle Contact Representations and Duality. In: Graph Drawing 18th International Symposium, GD 2010, September 2124, 2010 , pp. 262273(Official URL: http://dx.doi.org/10.1007/9783642184697_24). Full text not available from this repository.
Official URL: http://dx.doi.org/10.1007/9783642184697_24
AbstractA 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 primaldual 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 3connected planar map admits a primaldual 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 onetoone correspondence with generalized Schnyder woods defined by Felsner for 3connected planar maps.
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