83 research outputs found
Relating Graph Thickness to Planar Layers and Bend Complexity
The thickness of a graph with vertices is the minimum number of
planar subgraphs of whose union is . A polyline drawing of in
is a drawing of , where each vertex is mapped to a
point and each edge is mapped to a polygonal chain. Bend and layer complexities
are two important aesthetics of such a drawing. The bend complexity of
is the maximum number of bends per edge in , and the layer complexity
of is the minimum integer such that the set of polygonal chains in
can be partitioned into disjoint sets, where each set corresponds
to a planar polyline drawing. Let be a graph of thickness . By
F\'{a}ry's theorem, if , then can be drawn on a single layer with bend
complexity . A few extensions to higher thickness are known, e.g., if
(resp., ), then can be drawn on layers with bend complexity 2
(resp., ). However, allowing a higher number of layers may reduce the
bend complexity, e.g., complete graphs require layers to be drawn
using 0 bends per edge.
In this paper we present an elegant extension of F\'{a}ry's theorem to draw
graphs of thickness . We first prove that thickness- graphs can be
drawn on layers with bends per edge. We then develop another
technique to draw thickness- graphs on layers with bend complexity,
i.e., , where . Previously, the bend complexity was not known to be sublinear for
. Finally, we show that graphs with linear arboricity can be drawn on
layers with bend complexity .Comment: A preliminary version appeared at the 43rd International Colloquium
on Automata, Languages and Programming (ICALP 2016
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
Simplified Emanation Graphs: A Sparse Plane Spanner with Steiner Points
Emanation graphs of grade , introduced by Hamedmohseni, Rahmati, and
Mondal, are plane spanners made by shooting rays from each given
point, where the shorter rays stop the longer ones upon collision. The
collision points are the Steiner points of the spanner.
We introduce a method of simplification for emanation graphs of grade ,
which makes it a competent spanner for many possible use cases such as network
visualization and geometric routing. In particular, the simplification reduces
the number of Steiner points by half and also significantly decreases the total
number of edges, without increasing the spanning ratio. Exact methods of
simplification along with mathematical proofs on properties of the simplified
graph is provided.
We compare simplified emanation graphs against Shewchuk's constrained
Delaunay triangulations on both synthetic and real-life datasets. Our
experimental results reveal that the simplified emanation graphs outperform
constrained Delaunay triangulations in common quality measures (e.g., edge
count, angular resolution, average degree, total edge length) while maintain a
comparable spanning ratio and Steiner point count.Comment: A preliminary and shorter version of this paper was accepted in
SOFSEM 202
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