4 research outputs found
Aligned Drawings of Planar Graphs
Let be a graph that is topologically embedded in the plane and let
be an arrangement of pseudolines intersecting the drawing of .
An aligned drawing of and is a planar polyline drawing
of with an arrangement of lines so that and are
homeomorphic to and . We show that if is
stretchable and every edge either entirely lies on a pseudoline or it has
at most one intersection with , then and have a
straight-line aligned drawing. In order to prove this result, we strengthen a
result of Da Lozzo et al., and prove that a planar graph and a single
pseudoline have an aligned drawing with a prescribed convex
drawing of the outer face. We also study the less restrictive version of the
alignment problem with respect to one line, where only a set of vertices is
given and we need to determine whether they can be collinear. We show that the
problem is NP-complete but fixed-parameter tractable.Comment: Preliminary work appeared in the Proceedings of the 25th
International Symposium on Graph Drawing and Network Visualization (GD 2017
The Complexity of Drawing a Graph in a Polygonal Region
We prove that the following problem is complete for the existential theory of
the reals: Given a planar graph and a polygonal region, with some vertices of
the graph assigned to points on the boundary of the region, place the remaining
vertices to create a planar straight-line drawing of the graph inside the
region. This strengthens an NP-hardness result by Patrignani on extending
partial planar graph drawings. Our result is one of the first showing that a
problem of drawing planar graphs with straight-line edges is hard for the
existential theory of the reals. The complexity of the problem is open in the
case of a simply connected region.
We also show that, even for integer input coordinates, it is possible that
drawing a graph in a polygonal region requires some vertices to be placed at
irrational coordinates. By contrast, the coordinates are known to be bounded in
the special case of a convex region, or for drawing a path in any polygonal
region.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018