7,072 research outputs found
Planar Octilinear Drawings with One Bend Per Edge
In octilinear drawings of planar graphs, every edge is drawn as an
alternating sequence of horizontal, vertical and diagonal ()
line-segments. In this paper, we study octilinear drawings of low edge
complexity, i.e., with few bends per edge. A -planar graph is a planar graph
in which each vertex has degree less or equal to . In particular, we prove
that every 4-planar graph admits a planar octilinear drawing with at most one
bend per edge on an integer grid of size . For 5-planar
graphs, we prove that one bend per edge still suffices in order to construct
planar octilinear drawings, but in super-polynomial area. However, for 6-planar
graphs we give a class of graphs whose planar octilinear drawings require at
least two bends per edge
Outerplanar graph drawings with few slopes
We consider straight-line outerplanar drawings of outerplanar graphs in which
a small number of distinct edge slopes are used, that is, the segments
representing edges are parallel to a small number of directions. We prove that
edge slopes suffice for every outerplanar graph with maximum degree
. This improves on the previous bound of , which was
shown for planar partial 3-trees, a superclass of outerplanar graphs. The bound
is tight: for every there is an outerplanar graph with maximum
degree that requires at least distinct edge slopes in an
outerplanar straight-line drawing.Comment: Major revision of the whole pape
Optimal partisan districting on planar geographies
We show that optimal partisan districting in the plane with geographical constraints is an NP-complete problem
Decomposition of Geometric Set Systems and Graphs
We study two decomposition problems in combinatorial geometry. The first part
deals with the decomposition of multiple coverings of the plane. We say that a
planar set is cover-decomposable if there is a constant m such that any m-fold
covering of the plane with its translates is decomposable into two disjoint
coverings of the whole plane. Pach conjectured that every convex set is
cover-decomposable. We verify his conjecture for polygons. Moreover, if m is
large enough, we prove that any m-fold covering can even be decomposed into k
coverings. Then we show that the situation is exactly the opposite in 3
dimensions, for any polyhedron and any we construct an m-fold covering of
the space that is not decomposable. We also give constructions that show that
concave polygons are usually not cover-decomposable. We start the first part
with a detailed survey of all results on the cover-decomposability of polygons.
The second part investigates another geometric partition problem, related to
planar representation of graphs. The slope number of a graph G is the smallest
number s with the property that G has a straight-line drawing with edges of at
most s distinct slopes and with no bends. We examine the slope number of
bounded degree graphs. Our main results are that if the maximum degree is at
least 5, then the slope number tends to infinity as the number of vertices
grows but every graph with maximum degree at most 3 can be embedded with only
five slopes. We also prove that such an embedding exists for the related notion
called slope parameter. Finally, we study the planar slope number, defined only
for planar graphs as the smallest number s with the property that the graph has
a straight-line drawing in the plane without any crossings such that the edges
are segments of only s distinct slopes. We show that the planar slope number of
planar graphs with bounded degree is bounded.Comment: This is my PhD thesi
Bitangents of tropical plane quartic curves
We study smooth tropical plane quartic curves and show that they satisfy
certain properties analogous to (but also different from) smooth plane quartics
in algebraic geometry. For example, we show that every such curve admits either
infinitely many or exactly 7 bitangent lines. We also prove that a smooth
tropical plane quartic curve cannot be hyperelliptic.Comment: 13 pages, 9 figures. Minor revisions; accepted for publication in
Mathematische Zeitschrif
Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends
We study the following classes of beyond-planar graphs: 1-planar, IC-planar,
and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar,
and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every
edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs
of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two
pairs of crossing edges share two vertices. We study the relations of these
beyond-planar graph classes (beyond-planar graphs is a collective term for the
primary attempts to generalize the planar graphs) to right-angle crossing (RAC)
graphs that admit compact drawings on the grid with few bends. We present four
drawing algorithms that preserve the given embeddings. First, we show that
every -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . Finally, we make two
known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at
most one bend per edge and for drawing IC-planar RAC graphs straight-line
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