5 research outputs found
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
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
Upward planar drawings with two slopes
In an upward planar 2-slope drawing of a digraph, edges are drawn as
straight-line segments in the upward direction without crossings using only two
different slopes. We investigate whether a given upward planar digraph admits
such a drawing and, if so, how to construct it. For the fixed embedding
scenario, we give a simple characterisation and a linear-time construction by
adopting algorithms from orthogonal drawings. For the variable embedding
scenario, we describe a linear-time algorithm for single-source digraphs, a
quartic-time algorithm for series-parallel digraphs, and a fixed-parameter
tractable algorithm for general digraphs. For the latter two classes, we make
use of SPQR-trees and the notion of upward spirality. As an application of this
drawing style, we show how to draw an upward planar phylogenetic network with
two slopes such that all leaves lie on a horizontal line
Drawing Cubic Graphs with the Four Basic Slopes
We show that every cubic graph can be drawn in the plane with straight-line edges using only the four basic slopes, {0, π/4, π/2, 3π/4}. We also prove that four slopes have this property if and only if we can draw K4 with them