2011 | OriginalPaper | Chapter
On Collinear Sets in Straight-Line Drawings
Authors : Alexander Ravsky, Oleg Verbitsky
Published in: Graph-Theoretic Concepts in Computer Science
Publisher: Springer Berlin Heidelberg
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We consider straight-line drawings of a planar graph
G
with possible edge crossings. The
untangling problem
is to eliminate all edge crossings by moving as few vertices as possible to new positions. Let
fix
G
denote the maximum number of vertices that can be left fixed in the worst case among all drawings of
G
. In the
allocation problem
, we are given a planar graph
G
on
n
vertices together with an
n
-point set
X
in the plane and have to draw
G
without edge crossings so that as many vertices as possible are located in
X
. Let
fit
G
denote the maximum number of points fitting this purpose in the worst case among all
n
-point sets
X
. As
fix
G
≤
fit
G
, we are interested in upper bounds for the latter and lower bounds for the former parameter.
For any
ε
> 0, we construct an infinite sequence of graphs with
fit
G
=
O
(
n
σ
+
ε
), where
σ
< 0.99 is a known graph-theoretic constant, namely the shortness exponent for the class of cubic polyhedral graphs. On the other hand, we prove that
$fix G\ge\sqrt{n/30}$
for any graph
G
of tree-width at most 2. This extends the lower bound obtained by Goaoc et al. [
Discrete and Computational Geometry
42:542–569 (2009)] for outerplanar graphs. Our results are based on estimating the maximum number of vertices that can be put on a line in a straight-line crossing-free drawing of a given planar graph.