59,932 research outputs found
Experimental analysis of the accessibility of drawings with few segments
The visual complexity of a graph drawing is defined as the number of
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 two
collinear incident edges. We study the question if drawings with few segments
have a better aesthetic appeal and help the user to asses the underlying graph.
We design an experiment that investigates two different graph types (trees and
sparse graphs), three different layout algorithms for trees, and two different
layout algorithms for sparse graphs. We asked the users to give an aesthetic
ranking on the layouts and to perform a furthest-pair or shortest-path task on
the drawings.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Trees with Convex Faces and Optimal Angles
We consider drawings of trees in which all edges incident to leaves can be
extended to infinite rays without crossing, partitioning the plane into
infinite convex polygons. Among all such drawings we seek the one maximizing
the angular resolution of the drawing. We find linear time algorithms for
solving this problem, both for plane trees and for trees without a fixed
embedding. In any such drawing, the edge lengths may be set independently of
the angles, without crossing; we describe multiple strategies for setting these
lengths.Comment: 12 pages, 10 figures. To appear at 14th Int. Symp. Graph Drawing,
200
Experimental Evaluation of Book Drawing Algorithms
A -page book drawing of a graph consists of a linear ordering of
its vertices along a spine and an assignment of each edge to one of the
pages, which are half-planes bounded by the spine. In a book drawing, two edges
cross if and only if they are assigned to the same page and their vertices
alternate along the spine. Crossing minimization in a -page book drawing is
NP-hard, yet book drawings have multiple applications in visualization and
beyond. Therefore several heuristic book drawing algorithms exist, but there is
no broader comparative study on their relative performance. In this paper, we
propose a comprehensive benchmark set of challenging graph classes for book
drawing algorithms and provide an extensive experimental study of the
performance of existing book drawing algorithms.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
A full characterization of invariant embeddability of unimodular planar graphs
When can a unimodular random planar graph be drawn in the Euclidean or the
hyperbolic plane in a way that the distribution of the random drawing is
isometry-invariant? This question was answered for one-ended unimodular graphs
in earlier work of Benjamini and Tim\'ar, using the fact that such graphs
automatically have locally finite (simply connected) drawings into the plane.
For the case of graphs with multiple ends the question was left open. We
provide a graph theoretic characterization of graphs that have a locally finite
embedding into the plane. Then we prove that such graphs do have a locally
finite invariant embedding into the Euclidean or the hyperbolic plane,
depending on whether the graph is amenable or not.Comment: Added a proof for Theorem 9. 32 pages, 3 figure
On Visibility Representations of Non-planar Graphs
A rectangle visibility representation (RVR) of a graph consists of an
assignment of axis-aligned rectangles to vertices such that for every edge
there exists a horizontal or vertical line of sight between the rectangles
assigned to its endpoints. Testing whether a graph has an RVR is known to be
NP-hard. In this paper, we study the problem of finding an RVR under the
assumption that an embedding in the plane of the input graph is fixed and we
are looking for an RVR that reflects this embedding. We show that in this case
the problem can be solved in polynomial time for general embedded graphs and in
linear time for 1-plane graphs (i.e., embedded graphs having at most one
crossing per edge). The linear time algorithm uses a precise list of forbidden
configurations, which extends the set known for straight-line drawings of
1-plane graphs. These forbidden configurations can be tested for in linear
time, and so in linear time we can test whether a 1-plane graph has an RVR and
either compute such a representation or report a negative witness. Finally, we
discuss some extensions of our study to the case when the embedding is not
fixed but the RVR can have at most one crossing per edge
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