We define some Schnyder-type combinatorial structures on a class of planar triangulations of the pentagon which are closely related to 5-connected triangulations. The combinatorial structures have three incarnations defined in terms of orientations, corner-labelings, and woods respectively. The wood incarnation consists in 5 spanning trees crossing each other in an orderly fashion. Similarly as for Schnyder woods on triangulations, it induces, for each vertex, a partition of the inner triangles into face-connected regions (5~regions here). We show that the induced barycentric vertex-placement, where each vertex is at the barycenter of the 5 outer vertices with weights given by the number of faces in each region, yields a planar straight-line drawing.

翻译：本文在五边形的一类平面三角剖分（与5-连通三角剖分密切相关）上定义了一些Schnyder型组合结构。这些组合结构具有三种表现形式，分别基于定向、角标记和森林进行定义。其中森林表现形式由5棵以有序方式相互交叉的生成树组成。类似于三角剖分上的Schnyder森林，该结构为每个顶点诱导出一个将内三角形划分为面连通区域（此处为5个区域）的分割。我们证明，由此导出的重心顶点布局（每个顶点位于5个外顶点的重心位置，权重由各区域内的面数决定）能够生成平面直线型绘图。