29,623 research outputs found
Pole Dancing: 3D Morphs for Tree Drawings
We study the question whether a crossing-free 3D morph between two
straight-line drawings of an -vertex tree can be constructed consisting of a
small number of linear morphing steps. We look both at the case in which the
two given drawings are two-dimensional and at the one in which they are
three-dimensional. In the former setting we prove that a crossing-free 3D morph
always exists with steps, while for the latter steps
are always sufficient and sometimes necessary.Comment: Appears in the Proceedings of the 26th International Symposium on
Graph Drawing and Network Visualization (GD 2018
Expansive Motions and the Polytope of Pointed Pseudo-Triangulations
We introduce the polytope of pointed pseudo-triangulations of a point set in
the plane, defined as the polytope of infinitesimal expansive motions of the
points subject to certain constraints on the increase of their distances. Its
1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of
the point set and whose edges are flips of interior pseudo-triangulation edges.
For points in convex position we obtain a new realization of the
associahedron, i.e., a geometric representation of the set of triangulations of
an n-gon, or of the set of binary trees on n vertices, or of many other
combinatorial objects that are counted by the Catalan numbers. By considering
the 1-dimensional version of the polytope of constrained expansive motions we
obtain a second distinct realization of the associahedron as a perturbation of
the positive cell in a Coxeter arrangement.
Our methods produce as a by-product a new proof that every simple polygon or
polygonal arc in the plane has expansive motions, a key step in the proofs of
the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by
Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially
to the "Further remarks" in Section 5) + changed to final book format. This
version is to appear in "Discrete and Computational Geometry -- The
Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds),
series "Algorithms and Combinatorics", Springer Verlag, Berli
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
Polynomiality, Wall Crossings and Tropical Geometry of Rational Double Hurwitz Cycles
We study rational double Hurwitz cycles, i.e. loci of marked rational stable
curves admitting a map to the projective line with assigned ramification
profiles over two fixed branch points. Generalizing the phenomenon observed for
double Hurwitz numbers, such cycles are piecewise polynomial in the entries of
the special ramification; the chambers of polynomiality and wall crossings have
an explicit and "modular" description. A main goal of this paper is to
simultaneously carry out this investigation for the corresponding objects in
tropical geometry, underlining a precise combinatorial duality between
classical and tropical Hurwitz theory
Area Inequalities for Embedded Disks Spanning Unknotted Curves
We show that a smooth unknotted curve in R^3 satisfies an isoperimetric
inequality that bounds the area of an embedded disk spanning the curve in terms
of two parameters: the length L of the curve and the thickness r (maximal
radius of an embedded tubular neighborhood) of the curve. For fixed length, the
expression giving the upper bound on the area grows exponentially in 1/r^2. In
the direction of lower bounds, we give a sequence of length one curves with r
approaching 0 for which the area of any spanning disk is bounded from below by
a function that grows exponentially with 1/r. In particular, given any constant
A, there is a smooth, unknotted length one curve for which the area of a
smallest embedded spanning disk is greater than A.Comment: 31 pages, 5 figure
The Computational Complexity of Knot and Link Problems
We consider the problem of deciding whether a polygonal knot in 3-dimensional
Euclidean space is unknotted, capable of being continuously deformed without
self-intersection so that it lies in a plane. We show that this problem, {\sc
unknotting problem} is in {\bf NP}. We also consider the problem, {\sc
unknotting problem} of determining whether two or more such polygons can be
split, or continuously deformed without self-intersection so that they occupy
both sides of a plane without intersecting it. We show that it also is in NP.
Finally, we show that the problem of determining the genus of a polygonal knot
(a generalization of the problem of determining whether it is unknotted) is in
{\bf PSPACE}. We also give exponential worst-case running time bounds for
deterministic algorithms to solve each of these problems. These algorithms are
based on the use of normal surfaces and decision procedures due to W. Haken,
with recent extensions by W. Jaco and J. L. Tollefson.Comment: 32 pages, 1 figur
Weak hyperbolicity of cube complexes and quasi-arboreal groups
We examine a graph encoding the intersection of hyperplane carriers
in a CAT(0) cube complex . The main result is that is
quasi-isometric to a tree. This implies that a group acting properly and
cocompactly on is weakly hyperbolic relative to the hyperplane
stabilizers. Using disc diagram techniques and Wright's recent result on the
aymptotic dimension of CAT(0) cube complexes, we give a generalization of a
theorem of Bell and Dranishnikov on the finite asymptotic dimension of graphs
of asymptotically finite-dimensional groups. More precisely, we prove
asymptotic finite-dimensionality for finitely-generated groups acting on
finite-dimensional cube complexes with 0-cube stabilizers of uniformly bounded
asymptotic dimension. Finally, we apply contact graph techniques to prove a
cubical version of the flat plane theorem stated in terms of complete bipartite
subgraphs of .Comment: Corrections in Sections 2 and 4. Simplification in Section
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