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 \documentclass[12pt]{minimal}
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\begin{document}$\lceil \frac{15n}{16} \rceil$\end{document}. This improves the previous best bound of n-1. The drawing can be obtained in linear time. 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). This embedding can also be found in O(n) time.
493-504
chapters
minimum realizer
Schnyder’s realizers
realizers
Compact Visibility Representation and Straight-Line Grid Embedding of Plane Graphs
ordering trees
time
visibility representation
plane graph
plane graph G
grid
number
Compact Visibility Representation
cyclic faces
en
drawings
face
graph G
2022-01-01T19:12
vertices
chapter
straight-line grid
representation
compact drawings
https://doi.org/10.1007/978-3-540-45078-8_43
height
2003
false
properties
N-1
linear time
grid embedding
embedding
Straight-Line Grid Embedding
Δ0
https://scigraph.springernature.com/explorer/license/
style
trees
canonical ordering tree
2003-01-01
respect
graph
Zhang
Huaming
Springer Nature
Michiel
Smid
978-3-540-45078-8
978-3-540-40545-0
Algorithms and Data Structures
Sack
Jörg-Rüdiger
Mathematical Sciences
Xin
He
doi
10.1007/978-3-540-45078-8_43
dimensions_id
pub.1007042305
Pure Mathematics
Springer Nature - SN SciGraph project
Frank
Dehne
Department of Computer Science and Engineering, SUNY at Buffalo, 14260, Buffalo, NY, USA
Department of Computer Science and Engineering, SUNY at Buffalo, 14260, Buffalo, NY, USA