1,056 research outputs found
Orthogonal Graph Drawing with Inflexible Edges
We consider the problem of creating plane orthogonal drawings of 4-planar
graphs (planar graphs with maximum degree 4) with constraints on the number of
bends per edge. More precisely, we have a flexibility function assigning to
each edge a natural number , its flexibility. The problem
FlexDraw asks whether there exists an orthogonal drawing such that each edge
has at most bends. It is known that FlexDraw is NP-hard
if for every edge . On the other hand, FlexDraw can
be solved efficiently if and is trivial if
for every edge .
To close the gap between the NP-hardness for and the
efficient algorithm for , we investigate the
computational complexity of FlexDraw in case only few edges are inflexible
(i.e., have flexibility~). We show that for any FlexDraw
is NP-complete for instances with inflexible edges with
pairwise distance (including the case where they
induce a matching). On the other hand, we give an FPT-algorithm with running
time , where
is the time necessary to compute a maximum flow in a planar flow network with
multiple sources and sinks, and is the number of inflexible edges having at
least one endpoint of degree 4.Comment: 23 pages, 5 figure
Transforming planar graph drawings while maintaining height
There are numerous styles of planar graph drawings, notably straight-line
drawings, poly-line drawings, orthogonal graph drawings and visibility
representations. In this note, we show that many of these drawings can be
transformed from one style to another without changing the height of the
drawing. We then give some applications of these transformations
Simultaneous Orthogonal Planarity
We introduce and study the problem: Given planar
graphs each with maximum degree 4 and the same vertex set, do they admit an
OrthoSEFE, that is, is there an assignment of the vertices to grid points and
of the edges to paths on the grid such that the same edges in distinct graphs
are assigned the same path and such that the assignment induces a planar
orthogonal drawing of each of the graphs?
We show that the problem is NP-complete for even if the shared
graph is a Hamiltonian cycle and has sunflower intersection and for
even if the shared graph consists of a cycle and of isolated vertices. Whereas
the problem is polynomial-time solvable for when the union graph has
maximum degree five and the shared graph is biconnected. Further, when the
shared graph is biconnected and has sunflower intersection, we show that every
positive instance has an OrthoSEFE with at most three bends per edge.Comment: Appears in the Proceedings of the 24th International Symposium on
Graph Drawing and Network Visualization (GD 2016
Planar L-Drawings of Directed Graphs
We study planar drawings of directed graphs in the L-drawing standard. We
provide necessary conditions for the existence of these drawings and show that
testing for the existence of a planar L-drawing is an NP-complete problem.
Motivated by this result, we focus on upward-planar L-drawings. We show that
directed st-graphs admitting an upward- (resp. upward-rightward-) planar
L-drawing are exactly those admitting a bitonic (resp. monotonically
increasing) st-ordering. We give a linear-time algorithm that computes a
bitonic (resp. monotonically increasing) st-ordering of a planar st-graph or
reports that there exists none.Comment: Appears in the Proceedings of the 25th International Symposium on
Graph Drawing and Network Visualization (GD 2017
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
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