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 \emph{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 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 rooted at v∗ such
that each edge not incident to v∗ is used in opposite directions by two
trees. 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, the dual of even Schnyder
decompositions yields (planar) orthogonal and straight-line drawing algorithms.
For a 4-regular plane graph G of mincut 4 with n vertices plus a marked
vertex v, the vertices of G\v are placed on a (n−1)×(n−1) grid according to a permutation pattern, and in the orthogonal drawing
each of the 2n−2 edges of G\v has exactly one bend. Embedding
also the marked vertex v is doable at the cost of two additional rows and
columns and 8 additional bends for the 4 edges incident to v. We propose a
further compaction step for the drawing algorithm and show that the obtained
grid-size is strongly concentrated around 25n/32×25n/32 for a uniformly
random instance with n vertices