151 research outputs found
The role of twins in computing planar supports of hypergraphs
A support or realization of a hypergraph is a graph on the same
vertex as such that for each hyperedge of it holds that its vertices
induce a connected subgraph of . The NP-hard problem of finding a planar}
support has applications in hypergraph drawing and network design. Previous
algorithms for the problem assume that twins}---pairs of vertices that are in
precisely the same hyperedges---can safely be removed from the input
hypergraph. We prove that this assumption is generally wrong, yet that the
number of twins necessary for a hypergraph to have a planar support only
depends on its number of hyperedges. We give an explicit upper bound on the
number of twins necessary for a hypergraph with hyperedges to have an
-outerplanar support, which depends only on and . Since all
additional twins can be safely removed, we obtain a linear-time algorithm for
computing -outerplanar supports for hypergraphs with hyperedges if
and are constant; in other words, the problem is fixed-parameter
linear-time solvable with respect to the parameters and
Simultaneous Embeddability of Two Partitions
We study the simultaneous embeddability of a pair of partitions of the same
underlying set into disjoint blocks. Each element of the set is mapped to a
point in the plane and each block of either of the two partitions is mapped to
a region that contains exactly those points that belong to the elements in the
block and that is bounded by a simple closed curve. We establish three main
classes of simultaneous embeddability (weak, strong, and full embeddability)
that differ by increasingly strict well-formedness conditions on how different
block regions are allowed to intersect. We show that these simultaneous
embeddability classes are closely related to different planarity concepts of
hypergraphs. For each embeddability class we give a full characterization. We
show that (i) every pair of partitions has a weak simultaneous embedding, (ii)
it is NP-complete to decide the existence of a strong simultaneous embedding,
and (iii) the existence of a full simultaneous embedding can be tested in
linear time.Comment: 17 pages, 7 figures, extended version of a paper to appear at GD 201
05191 Abstracts Collection -- Graph Drawing
From 08.05.05 to 13.05.05, the Dagstuhl Seminar 05191 ``Graph Drawing\u27\u27 was held
in the International Conference and Research Center (IBFI), Schloss Dagstuhl.
During the seminar, several participants presented their current
research, and ongoing work and open problems were discussed. Abstracts of
the presentations given during the seminar as well as abstracts of
seminar results and ideas are put together in this paper. The first section
describes the seminar topics and goals in general.
Links to extended abstracts or full papers are provided, if available
Dependent Random Choice
We describe a simple and yet surprisingly powerful probabilistic technique
which shows how to find in a dense graph a large subset of vertices in which
all (or almost all) small subsets have many common neighbors. Recently this
technique has had several striking applications to Extremal Graph Theory,
Ramsey Theory, Additive Combinatorics, and Combinatorial Geometry. In this
survey we discuss some of them.Comment: 32 page
- …