Schnyder Decompositions for Regular Plane Graphs and Application to Drawing

Abstract

Schnyder woods are decompositions of simple triangulations into three edge-disjoint spanning trees crossing each other in a specific way. In this article, we generalize the definition 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 d-angulation 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 of a fixed d-angulation of girth d has a natural structure of distributive lattice. We also study the dual of Schnyder decompositions which are defined on d-regular plane graphs of mincut d with a distinguished vertex v ^∗: these are sets of d spanning trees rooted at v ^∗ crossing each other in a specific way and such that each edge not incident to v ^∗ is used by two trees in opposite directions. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d=4, these correspond to well-studied structures on simple quadrangulations (2-orientations and partitions into 2 spanning trees).

In the case d=4, we obtain straight-line and orthogonal planar drawing algorithms by using the dual of even Schnyder decompositions. For a 4-regular plane graph G of mincut 4 with a distinguished vertex v ^∗ and n−1 other vertices, our algorithms places the vertices of G\v ^∗ on a (n−2)×(n−2) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−4 edges of G\v ^∗ has exactly one bend. The vertex v ^∗ can be embedded at the cost of 3 additional rows and columns, and 8 additional bends.

We also describe a further compaction step for the drawing algorithms and show that the obtained grid-size is strongly concentrated around 25n/32×25n/32 for a uniformly random instance with n vertices.