15 research outputs found
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
Recognizing and Drawing IC-planar Graphs
IC-planar graphs are those graphs that admit a drawing where no two crossed
edges share an end-vertex and each edge is crossed at most once. They are a
proper subfamily of the 1-planar graphs. Given an embedded IC-planar graph
with vertices, we present an -time algorithm that computes a
straight-line drawing of in quadratic area, and an -time algorithm
that computes a straight-line drawing of with right-angle crossings in
exponential area. Both these area requirements are worst-case optimal. We also
show that it is NP-complete to test IC-planarity both in the general case and
in the case in which a rotation system is fixed for the input graph.
Furthermore, we describe a polynomial-time algorithm to test whether a set of
matching edges can be added to a triangulated planar graph such that the
resulting graph is IC-planar
A Heuristic Approach Towards Drawings of Graphs with High Crossing Resolution
The crossing resolution of a non-planar drawing of a graph is the value of
the minimum angle formed by any pair of crossing edges. Recent experiments have
shown that the larger the crossing resolution is, the easier it is to read and
interpret a drawing of a graph. However, maximizing the crossing resolution
turns out to be an NP-hard problem in general and only heuristic algorithms are
known that are mainly based on appropriately adjusting force-directed
algorithms.
In this paper, we propose a new heuristic algorithm for the crossing
resolution maximization problem and we experimentally compare it against the
known approaches from the literature. Our experimental evaluation indicates
that the new heuristic produces drawings with better crossing resolution, but
this comes at the cost of slightly higher aspect ratio, especially when the
input graph is large.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
1-Fan-bundle-planar drawings of graphs
Edge bundling is an important concept heavily used for graph visualization purposes. To enable the comparison with other established near-planarity models in graph drawing, we formulate a new edge-bundling model which is inspired by the recently introduced fan-planar graphs. In particular, we restrict the bundling to the endsegments of the edges. Similarly to 1-planarity, we call our model 1-fan-bundle-planarity, as we allow at most one crossing per bundle.
For the two variants where we allow either one or, more naturally, both endsegments of each edge to be part of bundles, we present edge density results and consider various recognition questions, not only for general graphs, but also for the outer and 2-layer variants. We conclude with a series of challenging questions
The QuaSEFE Problem
ISSN:0302-9743ISSN:1611-334