Schnyder decompositions for regular plane graphs and application to drawing

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to d-angulations (plane graphs with faces of degree d) for all d 3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d - 2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the dangulation is d. As in the case of Schnyder woods (d = 3), there are alternative formulations in terms of orientations ("fractional" orientations when d 5) and in terms of corner-labellings. Moreover, the set of Schnyder decompositions on a fixed d-angulation of girth d is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on d-regular plane graphs of mincut d rooted at a vertex v) are decompositions into d spanning trees...

• Type: Journal
• Year: 2010
• Where: CORR
• Authors: Olivier Bernardi, Éric Fusy
• Keywords: CORR 2010, Education, Plane Graph, Schnyder Decompositions, Schnyder Woods