Schnyder decompositions for regular plane graphs and application to drawing

Olivier Bernardi (1) , Eric Fusy (2)

1 MIT - Massachusetts Institute of Technology
2 LIX - Laboratoire d'informatique de l'École polytechnique [Palaiseau]

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 $d\geq 3$. 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 $d\geq 5$) 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 $G\backslash v$ are placed on a $(n-1) \times (n-1)$ grid according to a permutation pattern, and in the orthogonal drawing each of the $2n-2$ edges of $G\backslash v$ 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 $25n/32\times 25n/32$ for a uniformly random instance with $n$ vertices.

Domaines: Combinatoire [math.CO]; Algorithme et structure de données [cs.DS]; Mathématique discrète [cs.DM]

Olivier Bernardi, Eric Fusy. Schnyder decompositions for regular plane graphs and application to drawing. Algorithmica, 2012, 62, pp 1159-1197. ⟨hal-00502445v2⟩