3 research outputs found
Simultaneous Embeddings with Few Bends and Crossings
A simultaneous embedding with fixed edges (SEFE) of two planar graphs and
is a pair of plane drawings of and that coincide when restricted to
the common vertices and edges of and . We show that whenever and
admit a SEFE, they also admit a SEFE in which every edge is a polygonal curve
with few bends and every pair of edges has few crossings. Specifically: (1) if
and are trees then one bend per edge and four crossings per edge pair
suffice (and one bend per edge is sometimes necessary), (2) if is a planar
graph and is a tree then six bends per edge and eight crossings per edge
pair suffice, and (3) if and are planar graphs then six bends per edge
and sixteen crossings per edge pair suffice. Our results improve on a paper by
Grilli et al. (GD'14), which proves that nine bends per edge suffice, and on a
paper by Chan et al. (GD'14), which proves that twenty-four crossings per edge
pair suffice.Comment: Full version of the paper "Simultaneous Embeddings with Few Bends and
Crossings" accepted at GD '1
Relaxing the Constraints of Clustered Planarity
In a drawing of a clustered graph vertices and edges are drawn as points and
curves, respectively, while clusters are represented by simple closed regions.
A drawing of a clustered graph is c-planar if it has no edge-edge, edge-region,
or region-region crossings. Determining the complexity of testing whether a
clustered graph admits a c-planar drawing is a long-standing open problem in
the Graph Drawing research area. An obvious necessary condition for c-planarity
is the planarity of the graph underlying the clustered graph. However, such a
condition is not sufficient and the consequences on the problem due to the
requirement of not having edge-region and region-region crossings are not yet
fully understood.
In order to shed light on the c-planarity problem, we consider a relaxed
version of it, where some kinds of crossings (either edge-edge, edge-region, or
region-region) are allowed even if the underlying graph is planar. We
investigate the relationships among the minimum number of edge-edge,
edge-region, and region-region crossings for drawings of the same clustered
graph. Also, we consider drawings in which only crossings of one kind are
admitted. In this setting, we prove that drawings with only edge-edge or with
only edge-region crossings always exist, while drawings with only region-region
crossings may not. Further, we provide upper and lower bounds for the number of
such crossings. Finally, we give a polynomial-time algorithm to test whether a
drawing with only region-region crossings exist for biconnected graphs, hence
identifying a first non-trivial necessary condition for c-planarity that can be
tested in polynomial time for a noticeable class of graphs