2,126 research outputs found
Extending Upward Planar Graph Drawings
In this paper we study the computational complexity of the Upward Planarity
Extension problem, which takes in input an upward planar drawing of
a subgraph of a directed graph and asks whether can be
extended to an upward planar drawing of . Our study fits into the line of
research on the extensibility of partial representations, which has recently
become a mainstream in Graph Drawing.
We show the following results.
First, we prove that the Upward Planarity Extension problem is NP-complete,
even if has a prescribed upward embedding, the vertex set of coincides
with the one of , and contains no edge.
Second, we show that the Upward Planarity Extension problem can be solved in
time if is an -vertex upward planar -graph. This
result improves upon a known -time algorithm, which however applies to
all -vertex single-source upward planar graphs.
Finally, we show how to solve in polynomial time a surprisingly difficult
version of the Upward Planarity Extension problem, in which is a directed
path or cycle with a prescribed upward embedding, contains no edges, and no
two vertices share the same -coordinate in
A Note on Plus-Contacts, Rectangular Duals, and Box-Orthogonal Drawings
A plus-contact representation of a planar graph is called -balanced if
for every plus shape , the number of other plus shapes incident to each
arm of is at most , where is the maximum degree
of . Although small values of have been achieved for a few subclasses of
planar graphs (e.g., - and -trees), it is unknown whether -balanced
representations with exist for arbitrary planar graphs.
In this paper we compute -balanced plus-contact representations for
all planar graphs that admit a rectangular dual. Our result implies that any
graph with a rectangular dual has a 1-bend box-orthogonal drawings such that
for each vertex , the box representing is a square of side length
.Comment: A poster related to this research appeared at the 25th International
Symposium on Graph Drawing & Network Visualization (GD 2017
Straight-line Drawability of a Planar Graph Plus an Edge
We investigate straight-line drawings of topological graphs that consist of a
planar graph plus one edge, also called almost-planar graphs. We present a
characterization of such graphs that admit a straight-line drawing. The
characterization enables a linear-time testing algorithm to determine whether
an almost-planar graph admits a straight-line drawing, and a linear-time
drawing algorithm that constructs such a drawing, if it exists. We also show
that some almost-planar graphs require exponential area for a straight-line
drawing
Knuthian Drawings of Series-Parallel Flowcharts
Inspired by a classic paper by Knuth, we revisit the problem of drawing
flowcharts of loop-free algorithms, that is, degree-three series-parallel
digraphs. Our drawing algorithms show that it is possible to produce Knuthian
drawings of degree-three series-parallel digraphs with good aspect ratios and
small numbers of edge bends.Comment: Full versio
Drawing Trees with Perfect Angular Resolution and Polynomial Area
We study methods for drawing trees with perfect angular resolution, i.e.,
with angles at each node v equal to 2{\pi}/d(v). We show:
1. Any unordered tree has a crossing-free straight-line drawing with perfect
angular resolution and polynomial area.
2. There are ordered trees that require exponential area for any
crossing-free straight-line drawing having perfect angular resolution.
3. Any ordered tree has a crossing-free Lombardi-style drawing (where each
edge is represented by a circular arc) with perfect angular resolution and
polynomial area. Thus, our results explore what is achievable with
straight-line drawings and what more is achievable with Lombardi-style
drawings, with respect to drawings of trees with perfect angular resolution.Comment: 30 pages, 17 figure
Notes on large angle crossing graphs
A graph G is an a-angle crossing (aAC) graph if every pair of crossing edges
in G intersect at an angle of at least a. The concept of right angle crossing
(RAC) graphs (a=Pi/2) was recently introduced by Didimo et. al. It was shown
that any RAC graph with n vertices has at most 4n-10 edges and that there are
infinitely many values of n for which there exists a RAC graph with n vertices
and 4n-10 edges. In this paper, we give upper and lower bounds for the number
of edges in aAC graphs for all 0 < a < Pi/2
The Partial Visibility Representation Extension Problem
For a graph , a function is called a \emph{bar visibility
representation} of when for each vertex , is a
horizontal line segment (\emph{bar}) and iff there is an
unobstructed, vertical, -wide line of sight between and
. Graphs admitting such representations are well understood (via
simple characterizations) and recognizable in linear time. For a directed graph
, a bar visibility representation of , additionally, puts the bar
strictly below the bar for each directed edge of
. We study a generalization of the recognition problem where a function
defined on a subset of is given and the question is whether
there is a bar visibility representation of with for every . We show that for undirected graphs this problem
together with closely related problems are \NP-complete, but for certain cases
involving directed graphs it is solvable in polynomial time.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
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
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