12,238 research outputs found
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
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
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
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
The Flip Diameter of Rectangulations and Convex Subdivisions
We study the configuration space of rectangulations and convex subdivisions
of points in the plane. It is shown that a sequence of
elementary flip and rotate operations can transform any rectangulation to any
other rectangulation on the same set of points. This bound is the best
possible for some point sets, while operations are sufficient and
necessary for others. Some of our bounds generalize to convex subdivisions of
points in the plane.Comment: 17 pages, 12 figures, an extended abstract has been presented at
LATIN 201
Computing k-Modal Embeddings of Planar Digraphs
Given a planar digraph G and a positive even integer k, an embedding of G in the plane is k-modal, if every vertex of G is incident to at most k pairs of consecutive edges with opposite orientations, i.e., the incoming and the outgoing edges at each vertex are grouped by the embedding into at most k sets of consecutive edges with the same orientation. In this paper, we study the k-Modality problem, which asks for the existence of a k-modal embedding of a planar digraph. This combinatorial problem is at the very core of a variety of constrained embedding questions for planar digraphs and flat clustered networks.
First, since the 2-Modality problem can be easily solved in linear time, we consider the general k-Modality problem for any value of k>2 and show that the problem is NP-complete for planar digraphs of maximum degree Delta <= k+3. We relate its computational complexity to that of two notions of planarity for flat clustered networks: Planar Intersection-Link and Planar NodeTrix representations. This allows us to answer in the strongest possible way an open question by Di Giacomo [https://doi.org/10.1007/978-3-319-73915-1_37], concerning the complexity of constructing planar NodeTrix representations of flat clustered networks with small clusters, and to address a research question by Angelini et al. [https://doi.org/10.7155/jgaa.00437], concerning intersection-link representations based on geometric objects that determine complex arrangements. On the positive side, we provide a simple FPT algorithm for partial 2-trees of arbitrary degree, whose running time is exponential in k and linear in the input size. Second, motivated by the recently-introduced planar L-drawings of planar digraphs [https://doi.org/10.1007/978-3-319-73915-1_36], which require the computation of a 4-modal embedding, we focus our attention on k=4. On the algorithmic side, we show a complexity dichotomy for the 4-Modality problem with respect to Delta, by providing a linear-time algorithm for planar digraphs with Delta <= 6. This algorithmic result is based on decomposing the input digraph into its blocks via BC-trees and each of these blocks into its triconnected components via SPQR-trees. In particular, we are able to show that the constraints imposed on the embedding by the rigid triconnected components can be tackled by means of a small set of reduction rules and discover that the algorithmic core of the problem lies in special instances of NAESAT, which we prove to be always NAE-satisfiable - a result of independent interest that improves on Porschen et al. [https://doi.org/10.1007/978-3-540-24605-3_14]. Finally, on the combinatorial side, we consider outerplanar digraphs and show that any such a digraph always admits a k-modal embedding with k=4 and that this value of k is best possible for the digraphs in this family
Interactive visualization of information hierarchies and applications on the web
The visualization of information hierarchies is concerned with the presentation of abstract hierarchical information about relationships between various entities. It has many applications in diverse domains such as software engineering, information systems, biology, and chemistry. Information hierarchies are typically modeled by an abstract tree, where vertices are entities and edges represent relationships between entities. The aim of visualizing tree drawings is to automatically produce drawings of trees which clearly reflect the relationships of the information hierarchy.
This thesis is primarily concerned with problems related to the automatic generation of area-efficient grid drawings of trees, interactively visualizing information hierarchies, and applying our techniques on Web data.
The main achievements of this thesis include:
1. An experimental study on algorithms that produce planar straight-line grid drawings of binary trees,
2. An experimental study that shows the algorithm for producing planar straight-line grid drawings of degree-d trees with n nodes with optimal linear area and with user-defined arbitrary aspect ratio, works well in practice,
3. A rings-based technique for interactively visualizing information hierarchies, in real-time,
4. A survey of Web visualization systems developed to address the lost in cyberspace problem,
5. A separation-based Web visualization system that we present as a viable solution to the lost in cyberspace problem,
6. A rings-based Web visualization system that we propose as a solution to the lost in cyberspace problem
SPQR-tree-like embedding representation for level planarity
An SPQR-tree is a data structure that efficiently represents all planar embeddings of a connected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints. We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give an algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using LP-trees as a drop-in replacement for SPQR-trees
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
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