24 research outputs found
Sweeps, arrangements and signotopes
AbstractSweeping is an important algorithmic tool in geometry. In the first part of this paper we define sweeps of arrangements and use the âSweeping Lemmaâ to show that Euclidean arrangements of pseudolines can be represented by wiring diagrams and zonotopal tilings. In the second part we introduce a further representation for Euclidean arrangements of pseudolines. This representation records an âorientationâ for each triple of lines. It turns out that a âtriple orientationâ corresponds to an arrangement exactly if it obeys a generalized transitivity law. Moreover, the âtriple orientationsâ carry a natural order relation which induces an order relation on arrangements. A closer look on the combinatorics behind this leads to a series of signotope orders closely related to higher Bruhat orders. We investigate the structure of higher Bruhat orders and give new purely combinatorial proofs for the main structural properties. Finally, we reconnect the combinatorics of the second part to geometry. In particular, we show that the maximum chains in the higher Bruhat orders correspond to sweeps
Flip Graph Connectivity for Arrangements of Pseudolines and Pseudocircles
Flip graphs of combinatorial and geometric objects are at the heart of many
deep structural insights and connections between different branches of discrete
mathematics and computer science. They also provide a natural framework for the
study of reconfiguration problems. We study flip graphs of arrangements of
pseudolines and of arrangements of pseudocircles, which are combinatorial
generalizations of lines and circles, respectively. In both cases we consider
triangle flips as local transformation and prove conjectures regarding their
connectivity.
In the case of pseudolines we show that the connectivity of the flip
graph equals its minimum degree, which is exactly . For the proof we
introduce the class of shellable line arrangements, which serve as reference
objects for the construction of disjoint paths. In fact, shellable arrangements
are elements of a flip graph of line arrangements which are vertices of a
polytope (Felsner and Ziegler; DM 241 (2001), 301--312). This polytope forms a
cluster of good connectivity in the flip graph of pseudolines. In the case of
pseudocircles we show that triangle flips induce a connected flip graph on
\emph{intersecting} arrangements and also on cylindrical intersecting
arrangements. The result for cylindrical arrangements is used in the proof for
intersecting arrangements. We also show that in both settings the diameter of
the flip graph is in . Our constructions make essential use of
variants of the sweeping lemma for pseudocircle arrangements (Snoeyink and
Hershberger; Proc.\ SoCG 1989: 354--363). We finally study cylindrical
arrangements in their own right and provide new combinatorial characterizations
of this class
Cubic Partial Cubes from Simplicial Arrangements
We show how to construct a cubic partial cube from any simplicial arrangement
of lines or pseudolines in the projective plane. As a consequence, we find nine
new infinite families of cubic partial cubes as well as many sporadic examples.Comment: 11 pages, 10 figure