Schnyder woods are defined for plane triangulations as special partitions of inner edges into three trees that cross each other in an orderly fashion. They are the cornerstone of Schnyder’s famous barycentric embedding algorithm.
In this talk, I will present some Schnyder-type combinatorial structures, called 5c-woods, on a class of plane triangulations of the pentagon satisfying some connectedness constraints. These structures have three incarnations, as 5-tuples of trees, corner labelings and orientations of some associated graphs. I will provide the existence result and discuss their connections to other previously known structures.
The wood incarnation of 5c-woods associates to each inner vertex a partition of the inner faces into five face-connected regions, which in turn induces Schnyder-type barycentric coordinates for the vertex. I will show that if one fixes the positions of the five outer vertices properly, then this vertex-placement rule produces a planar straight-line drawing.
This is a joint work with Olivier Bernardi and Eric Fusy.