Schnyder Decompositions for Regular Plane Graphs and Application to Drawing
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...
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| Vydáno v: | Algorithmica Ročník 62; číslo 3-4; s. 1159 - 1197 |
|---|---|
| Hlavní autoři: | , |
| Médium: | Journal Article |
| Jazyk: | angličtina |
| Vydáno: |
New York
Springer-Verlag
01.04.2012
Springer Springer Verlag |
| Témata: | |
| ISSN: | 0178-4617, 1432-0541 |
| On-line přístup: | Získat plný text |
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| Shrnutí: | 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 2
n
−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 25
n
/32×25
n
/32 for a uniformly random instance with
n
vertices. |
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| ISSN: | 0178-4617 1432-0541 |
| DOI: | 10.1007/s00453-011-9514-5 |