670 research outputs found
Triangulations without pointed spanning trees
Problem 50 in the Open Problems Project asks whether any triangulation on a point set in the plane contains a pointed spanning tree as a subgraph. We provide a counterexample. As a consequence we show that there exist
triangulations which require a linear number of edge flips to become Hamiltonian.Acciones Integradas 2003-2004Austrian Fonds zur Förderung der Wissenschaftlichen Forschun
Maximizing Maximal Angles for Plane Straight-Line Graphs
Let be a plane straight-line graph on a finite point set
in general position. The incident angles of a vertex
of are the angles between any two edges of that appear consecutively in
the circular order of the edges incident to .
A plane straight-line graph is called -open if each vertex has an
incident angle of size at least . In this paper we study the following
type of question: What is the maximum angle such that for any finite set
of points in general position we can find a graph from a certain
class of graphs on that is -open? In particular, we consider the
classes of triangulations, spanning trees, and paths on and give tight
bounds in most cases.Comment: 15 pages, 14 figures. Apart of minor corrections, some proofs that
were omitted in the previous version are now include
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
The greedy flip tree of a subword complex
We describe a canonical spanning tree of the ridge graph of a subword complex
on a finite Coxeter group. It is based on properties of greedy facets in
subword complexes, defined and studied in this paper. Searching this tree
yields an enumeration scheme for the facets of the subword complex. This
algorithm extends the greedy flip algorithm for pointed pseudotriangulations of
points or convex bodies in the plane.Comment: 14 pages, 10 figures; various corrections (in particular deletion of
Section 4 which contained a serious mistake pointed out by an anonymous
referee). This paper is subsumed by our joint results with Christian Stump on
"EL-labelings and canonical spanning trees for subword complexes"
(http://arxiv.org/abs/1210.1435) and will therefore not be publishe
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