65 research outputs found
Outerplanar graph drawings with few slopes
We consider straight-line outerplanar drawings of outerplanar graphs in which
a small number of distinct edge slopes are used, that is, the segments
representing edges are parallel to a small number of directions. We prove that
edge slopes suffice for every outerplanar graph with maximum degree
. This improves on the previous bound of , which was
shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound
is tight: for every there is an outerplanar graph with maximum
degree that requires at least distinct edge slopes in an
outerplanar straight-line drawing.Comment: Major revision of the whole pape
Drawings of Planar Graphs with Few Slopes and Segments
We study straight-line drawings of planar graphs with few segments and few
slopes. Optimal results are obtained for all trees. Tight bounds are obtained
for outerplanar graphs, 2-trees, and planar 3-trees. We prove that every
3-connected plane graph on vertices has a plane drawing with at most
segments and at most slopes. We prove that every cubic
3-connected plane graph has a plane drawing with three slopes (and three bends
on the outerface). In a companion paper, drawings of non-planar graphs with few
slopes are also considered.Comment: This paper is submitted to a journal. A preliminary version appeared
as "Really Straight Graph Drawings" in the Graph Drawing 2004 conference. See
http://arxiv.org/math/0606446 for a companion pape
Drawing Planar Graphs with Few Geometric Primitives
We define the \emph{visual complexity} of a plane graph drawing to be the
number of basic geometric objects needed to represent all its edges. In
particular, one object may represent multiple edges (e.g., one needs only one
line segment to draw a path with an arbitrary number of edges). Let denote
the number of vertices of a graph. We show that trees can be drawn with
straight-line segments on a polynomial grid, and with straight-line
segments on a quasi-polynomial grid. Further, we present an algorithm for
drawing planar 3-trees with segments on an
grid. This algorithm can also be used with a small modification to draw maximal
outerplanar graphs with edges on an grid. We also
study the problem of drawing maximal planar graphs with circular arcs and
provide an algorithm to draw such graphs using only arcs. This is
significantly smaller than the lower bound of for line segments for a
nontrivial graph class.Comment: Appeared at Proc. 43rd International Workshop on Graph-Theoretic
Concepts in Computer Science (WG 2017
Convex-Arc Drawings of Pseudolines
A weak pseudoline arrangement is a topological generalization of a line
arrangement, consisting of curves topologically equivalent to lines that cross
each other at most once. We consider arrangements that are outerplanar---each
crossing is incident to an unbounded face---and simple---each crossing point is
the crossing of only two curves. We show that these arrangements can be
represented by chords of a circle, by convex polygonal chains with only two
bends, or by hyperbolic lines. Simple but non-outerplanar arrangements
(non-weak) can be represented by convex polygonal chains or convex smooth
curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was
made as a poster at the 21st International Symposium on Graph Drawing,
Bordeaux, France, September 2013, and published in Lecture Notes in Computer
Science 8242, Springer, 2013, pp. 522--52
Planar Drawings with Few Slopes of Halin Graphs and Nested Pseudotrees
The of a planar graph is the
minimum number of edge slopes in a planar straight-line drawing of . It is
known that for every planar graph of maximum
degree . This upper bound has been improved to if has
treewidth three, and to if has treewidth two. In this paper we
prove when is a Halin graph, and thus has
treewidth three. Furthermore, we present the first polynomial upper bound on
the planar slope number for a family of graphs having treewidth four. Namely we
show that slopes suffice for nested pseudotrees.Comment: Extended version of "Planar Drawings with Few Slopes of Halin Graphs
and Nested Pseudotrees" appeared in the Proceedings of the 17th Algorithms
and Data Structures Symposium (WADS 2021
On Upward Drawings of Trees on a Given Grid
Computing a minimum-area planar straight-line drawing of a graph is known to
be NP-hard for planar graphs, even when restricted to outerplanar graphs.
However, the complexity question is open for trees. Only a few hardness results
are known for straight-line drawings of trees under various restrictions such
as edge length or slope constraints. On the other hand, there exist
polynomial-time algorithms for computing minimum-width (resp., minimum-height)
upward drawings of trees, where the height (resp., width) is unbounded.
In this paper we take a major step in understanding the complexity of the
area minimization problem for strictly-upward drawings of trees, which is one
of the most common styles for drawing rooted trees. We prove that given a
rooted tree and a grid, it is NP-hard to decide whether
admits a strictly-upward (unordered) drawing in the given grid.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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