Triangle Contact Representations and Duality

Résumé : 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.
Type de document : Communication dans un congrès
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http://hal-lirmm.ccsd.cnrs.fr/lirmm-00620728
Contributeur : Lévêque Benjamin <>
Soumis le : jeudi 8 septembre 2011 - 14:00:27
Dernière modification le : jeudi 8 septembre 2011 - 14:00:27

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Daniel Gonçalves, Benjamin Lévêque, Alexandre Pinlou. Triangle Contact Representations and Duality. GD'10: Graph Drawing, Sep 2011, Konstanz, Germany. pp.262-273, Lecture Notes in Computer Science. <10.1007/978-3-642-18469-7_24>. <lirmm-00620728>

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