On topological aspects of orientations
Discrete Mathematics
Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition
SODA '06 Proceedings of the seventeenth annual ACM-SIAM symposium on Discrete algorithm
Convex Drawings of 3-Connected Plane Graphs
Algorithmica
Barycentric systems and stretchability
Discrete Applied Mathematics
Schnyder Woods and Orthogonal Surfaces
Discrete & Computational Geometry
GD'10 Proceedings of the 18th international conference on Graph drawing
Optimal polygonal representation of planar graphs
LATIN'10 Proceedings of the 9th Latin American conference on Theoretical Informatics
Proportional contact representations of planar graphs
GD'11 Proceedings of the 19th international conference on Graph Drawing
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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.