34,253 research outputs found
Improved bounds for the crossing numbers of K_m,n and K_n
It has been long--conjectured that the crossing number cr(K_m,n) of the
complete bipartite graph K_m,n equals the Zarankiewicz Number Z(m,n):=
floor((m-1)/2) floor(m/2) floor((n-1)/2) floor(n/2). Another long--standing
conjecture states that the crossing number cr(K_n) of the complete graph K_n
equals Z(n):= floor(n/2) floor((n-1)/2) floor((n-2)/2) floor((n-3)/2)/4. In
this paper we show the following improved bounds on the asymptotic ratios of
these crossing numbers and their conjectured values:
(i) for each fixed m >= 9, lim_{n->infty} cr(K_m,n)/Z(m,n) >= 0.83m/(m-1);
(ii) lim_{n->infty} cr(K_n,n)/Z(n,n) >= 0.83; and
(iii) lim_{n->infty} cr(K_n)/Z(n) >= 0.83.
The previous best known lower bounds were 0.8m/(m-1), 0.8, and 0.8,
respectively. These improved bounds are obtained as a consequence of the new
bound cr(K_{7,n}) >= 2.1796n^2 - 4.5n. To obtain this improved lower bound for
cr(K_{7,n}), we use some elementary topological facts on drawings of K_{2,7} to
set up a quadratic program on 6! variables whose minimum p satisfies
cr(K_{7,n}) >= (p/2)n^2 - 4.5n, and then use state--of--the--art quadratic
optimization techniques combined with a bit of invariant theory of permutation
groups to show that p >= 4.3593.Comment: LaTeX, 18 pages, 2 figure
Drawing Binary Tanglegrams: An Experimental Evaluation
A binary tanglegram is a pair of binary trees whose leaf sets are in
one-to-one correspondence; matching leaves are connected by inter-tree edges.
For applications, for example in phylogenetics or software engineering, it is
required that the individual trees are drawn crossing-free. A natural
optimization problem, denoted tanglegram layout problem, is thus to minimize
the number of crossings between inter-tree edges.
The tanglegram layout problem is NP-hard and is currently considered both in
application domains and theory. In this paper we present an experimental
comparison of a recursive algorithm of Buchin et al., our variant of their
algorithm, the algorithm hierarchy sort of Holten and van Wijk, and an integer
quadratic program that yields optimal solutions.Comment: see
http://www.siam.org/proceedings/alenex/2009/alx09_011_nollenburgm.pd
Fixed parameter tractability of crossing minimization of almost-trees
We investigate exact crossing minimization for graphs that differ from trees
by a small number of additional edges, for several variants of the crossing
minimization problem. In particular, we provide fixed parameter tractable
algorithms for the 1-page book crossing number, the 2-page book crossing
number, and the minimum number of crossed edges in 1-page and 2-page book
drawings.Comment: Graph Drawing 201
Compact Drawings of 1-Planar Graphs with Right-Angle Crossings and Few Bends
We study the following classes of beyond-planar graphs: 1-planar, IC-planar,
and NIC-planar graphs. These are the graphs that admit a 1-planar, IC-planar,
and NIC-planar drawing, respectively. A drawing of a graph is 1-planar if every
edge is crossed at most once. A 1-planar drawing is IC-planar if no two pairs
of crossing edges share a vertex. A 1-planar drawing is NIC-planar if no two
pairs of crossing edges share two vertices. We study the relations of these
beyond-planar graph classes (beyond-planar graphs is a collective term for the
primary attempts to generalize the planar graphs) to right-angle crossing (RAC)
graphs that admit compact drawings on the grid with few bends. We present four
drawing algorithms that preserve the given embeddings. First, we show that
every -vertex NIC-planar graph admits a NIC-planar RAC drawing with at most
one bend per edge on a grid of size . Then, we show that
every -vertex 1-planar graph admits a 1-planar RAC drawing with at most two
bends per edge on a grid of size . Finally, we make two
known algorithms embedding-preserving; for drawing 1-planar RAC graphs with at
most one bend per edge and for drawing IC-planar RAC graphs straight-line
Analogies between the crossing number and the tangle crossing number
Tanglegrams are special graphs that consist of a pair of rooted binary trees
with the same number of leaves, and a perfect matching between the two
leaf-sets. These objects are of use in phylogenetics and are represented with
straightline drawings where the leaves of the two plane binary trees are on two
parallel lines and only the matching edges can cross. The tangle crossing
number of a tanglegram is the minimum crossing number over all such drawings
and is related to biologically relevant quantities, such as the number of times
a parasite switched hosts.
Our main results for tanglegrams which parallel known theorems for crossing
numbers are as follows. The removal of a single matching edge in a tanglegram
with leaves decreases the tangle crossing number by at most , and this
is sharp. Additionally, if is the maximum tangle crossing number of
a tanglegram with leaves, we prove
. Further,
we provide an algorithm for computing non-trivial lower bounds on the tangle
crossing number in time. This lower bound may be tight, even for
tanglegrams with tangle crossing number .Comment: 13 pages, 6 figure
Testing the Accuracy and Stability of Spectral Methods in Numerical Relativity
The accuracy and stability of the Caltech-Cornell pseudospectral code is
evaluated using the KST representation of the Einstein evolution equations. The
basic "Mexico City Tests" widely adopted by the numerical relativity community
are adapted here for codes based on spectral methods. Exponential convergence
of the spectral code is established, apparently limited only by numerical
roundoff error. A general expression for the growth of errors due to finite
machine precision is derived, and it is shown that this limit is achieved here
for the linear plane-wave test. All of these tests are found to be stable,
except for simulations of high amplitude gauge waves with nontrivial shift.Comment: Final version, as published in Phys. Rev. D; 13 pages, 16 figure
- âŚ