71,671 research outputs found
On Vertex- and Empty-Ply Proximity Drawings
We initiate the study of the vertex-ply of straight-line drawings, as a
relaxation of the recently introduced ply number. Consider the disks centered
at each vertex with radius equal to half the length of the longest edge
incident to the vertex. The vertex-ply of a drawing is determined by the vertex
covered by the maximum number of disks. The main motivation for considering
this relaxation is to relate the concept of ply to proximity drawings. In fact,
if we interpret the set of disks as proximity regions, a drawing with
vertex-ply number 1 can be seen as a weak proximity drawing, which we call
empty-ply drawing. We show non-trivial relationships between the ply number and
the vertex-ply number. Then, we focus on empty-ply drawings, proving some
properties and studying what classes of graphs admit such drawings. Finally, we
prove a lower bound on the ply and the vertex-ply of planar drawings.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
Empty Rectangles and Graph Dimension
We consider rectangle graphs whose edges are defined by pairs of points in
diagonally opposite corners of empty axis-aligned rectangles. The maximum
number of edges of such a graph on points is shown to be 1/4 n^2 +n -2.
This number also has other interpretations:
* It is the maximum number of edges of a graph of dimension
\bbetween{3}{4}, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\ol{\pi_1},\ol{\pi_2}.
* It is the number of 1-faces in a special Scarf complex.
The last of these interpretations allows to deduce the maximum number of
empty axis-aligned rectangles spanned by 4-element subsets of a set of
points. Moreover, it follows that the extremal point sets for the two problems
coincide.
We investigate the maximum number of of edges of a graph of dimension
, i.e., of a graph with a realizer of the form
\pi_1,\pi_2,\pi_3,\ol{\pi_3}. This maximum is shown to be .
Box graphs are defined as the 3-dimensional analog of rectangle graphs. The
maximum number of edges of such a graph on points is shown to be
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