4,254 research outputs found
On a Tree and a Path with no Geometric Simultaneous Embedding
Two graphs and admit a geometric simultaneous
embedding if there exists a set of points P and a bijection M: P -> V that
induce planar straight-line embeddings both for and for . While it
is known that two caterpillars always admit a geometric simultaneous embedding
and that two trees not always admit one, the question about a tree and a path
is still open and is often regarded as the most prominent open problem in this
area. We answer this question in the negative by providing a counterexample.
Additionally, since the counterexample uses disjoint edge sets for the two
graphs, we also negatively answer another open question, that is, whether it is
possible to simultaneously embed two edge-disjoint trees. As a final result, we
study the same problem when some constraints on the tree are imposed. Namely,
we show that a tree of depth 2 and a path always admit a geometric simultaneous
embedding. In fact, such a strong constraint is not so far from closing the gap
with the instances not admitting any solution, as the tree used in our
counterexample has depth 4.Comment: 42 pages, 33 figure
Polynomial Meshes: Computation and Approximation
We present the software package WAM, written in Matlab, that generates Weakly
Admissible Meshes and Discrete Extremal Sets of Fekete and Leja type, for 2d and 3d
polynomial least squares and interpolation on compact sets with various geometries.
Possible applications range from data fitting to high-order methods for PDEs
4-Holes in point sets
We consider a variant of a question of Erdos on the number of empty k-gons (k-holes) in a set of n points in the plane, where we allow the k-gons to be non-convex. We show bounds and structural results on maximizing and minimizing the number of general 4-holes, and maximizing the number of non-convex 4-holes. In particular, we show that for n >= 9, the maximum number of general 4-holes is ((pi)(4)); the minimum number of general 4-holes is at least 5/2 n(2) - circle minus(n); and the maximum number of non-convex 4-holes is at least 1/2 n(3) - circle minus(n(2) logn) and at most 1/2 n(3) - circle minus(n(2)). 2014 (c) Elsevier B.V. All rights reserved.Postprint (author’s final draft
Optimally fast incremental Manhattan plane embedding and planar tight span construction
We describe a data structure, a rectangular complex, that can be used to
represent hyperconvex metric spaces that have the same topology (although not
necessarily the same distance function) as subsets of the plane. We show how to
use this data structure to construct the tight span of a metric space given as
an n x n distance matrix, when the tight span is homeomorphic to a subset of
the plane, in time O(n^2), and to add a single point to a planar tight span in
time O(n). As an application of this construction, we show how to test whether
a given finite metric space embeds isometrically into the Manhattan plane in
time O(n^2), and add a single point to the space and re-test whether it has
such an embedding in time O(n).Comment: 39 pages, 15 figure
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