visibility
en
true
time
compact drawings
cyclic faces
number
plane triangulations
edge labeling
Canonical Ordering Trees and Their Applications in Graph Drawing
drawings
trees
article
triangulation
applications
labeling
ordering trees
regular edge labeling
linear time
properties
Abstract
We study the properties of Schnyder’s realizers and canonical ordering trees of plane graphs. Based on these newly
discovered properties, we obtain compact drawings of two styles for any plane graph G with n vertices. First, we show that G has a
visibility representation with height at most ⌈ 15n/16 ⌉. This improves the previous best bound
of (n - 1). Second, we show that every plane graph G has a straight-line grid
embedding on an (n - δ0 - 1) × (n - δ0 - 1) grid, where δ0 is the number of cyclic faces of G with respect to its
minimum realizer. This improves the previous best bound of (n - 1) × (n - 1).
We also study the properties of the regular edge labeling of 4-connected plane triangulation. Based on these
properties, we show that every such a graph has a canonical ordering tree with at most ⌈ (n + 1)/2 ⌉ leaves.
This improves the previously known bound of ⌊ (2n + 1)/3 ⌋.
We show that every 4-connected plane graph has a visibility
representation with height at most ⌈ 3n/4 ⌉.
All drawings discussed in this paper can be obtained in linear time.
2005-01-14
plane graph G
plane graph
style
2022-01-01T18:14
Schnyder’s realizers
straight-line grid
https://doi.org/10.1007/s00454-004-1154-y
leaves
graph drawing
respect
height
2005-01-14
representation
realizers
graph G
graph
Δ0
vertices
visibility representation
321-344
grid
articles
minimum realizer
paper
https://scigraph.springernature.com/explorer/license/
canonical ordering tree
face
He
Xin
Huaming
Zhang
pub.1021220162
dimensions_id
2
doi
10.1007/s00454-004-1154-y
Discrete & Computational Geometry
Springer Nature
0179-5376
1432-0444
Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
Department of Computer Science and Engineering, State University of New York at Buffalo, Buffalo, NY 14260, USA
Mathematical Sciences
Pure Mathematics
Springer Nature - SN SciGraph project
33