Allowable double-permutation sequences and double pseudoline arrangements

2008 ◽  
Vol 31 ◽  
pp. 167-168
Author(s):  
Ricky Pollack
Keyword(s):  
Pseudoline Arrangements

scholarly journals Symmetric Simplicial Pseudoline Arrangements

10.37236/737 ◽  
2008 ◽  
Vol 15 (1) ◽  
Author(s):  
Leah Wrenn Berman
Keyword(s):  
Projective Plane ◽  
Symmetry Classes ◽  
Geometric Symmetry ◽  
Pseudoline Arrangements

A simplicial arrangement of pseudolines is a collection of topological lines in the projective plane where each region that is formed is triangular. This paper refines and develops David Eppstein's notion of a kaleidoscope construction for symmetric pseudoline arrangements to construct and analyze several infinite families of simplicial pseudoline arrangements with high degrees of geometric symmetry. In particular, all simplicial pseudoline arrangements with the symmetries of a regular $k$-gon and three symmetry classes of pseudolines, consisting of the mirrors of the $k$-gon and two other symmetry classes, plus sometimes the line at infinity, are classified, and other interesting families (with more symmetry classes of pseudolines) are discussed.

scholarly journals Two-Dimensional Partial Cubes

10.37236/8934 ◽  
2020 ◽  
Vol 27 (3) ◽  
Author(s):  
Victor Chepoi ◽  
Kolja Knauer ◽  
Manon Philibert
Keyword(s):  
Cell Structure ◽  
Low Rank ◽  
Complete Graphs ◽  
Two Dimensional ◽  
Vc Dimension ◽  
Set Systems ◽  
Sample Compression ◽  
Vertex Set ◽  
Dimension 2 ◽  
Pseudoline Arrangements

We investigate the structure of two-dimensional partial cubes, i.e., of isometric subgraphs of hypercubes whose vertex set defines a set family of VC-dimension at most 2. Equivalently, those are the partial cubes which are not contractible to the 3-cube $Q_3$ (here contraction means contracting the edges corresponding to the same coordinate of the hypercube). We show that our graphs can be obtained from two types of combinatorial cells (gated cycles and gated full subdivisions of complete graphs) via amalgams. The cell structure of two-dimensional partial cubes enables us to establish a variety of results. In particular, we prove that all partial cubes of VC-dimension 2 can be extended to ample aka lopsided partial cubes of VC-dimension 2, yielding that the set families defined by such graphs satisfy the sample compression conjecture by Littlestone and Warmuth (1986) in a strong sense. The latter is a central conjecture of the area of computational machine learning, that is far from being solved even for general set systems of VC-dimension 2. Moreover, we point out relations to tope graphs of COMs of low rank and region graphs of pseudoline arrangements.

Polynomial realization of pseudoline arrangements

1985 ◽  
Vol 38 (6) ◽  
pp. 725-732 ◽  
Author(s):  
Jacob E. Goodman ◽  
Richard Pollack
Keyword(s):  
Pseudoline Arrangements

scholarly journals A Pseudoline Counterexample to the Strong Dirac Conjecture

10.37236/4015 ◽  
2014 ◽  
Vol 21 (2) ◽  
Author(s):  
Ben Lund ◽  
George B. Purdy ◽  
Justin W. Smith
Keyword(s):  
Infinite Family ◽  
Open Problems ◽  
Dirac Conjecture ◽  
Pseudoline Arrangements

We demonstrate an infinite family of pseudoline arrangements, in which an arrangement of $n$ pseudolines has no member incident to more than $4n/9$ points of intersection. This shows the "Strong Dirac" conjecture to be false for pseudolines.We also raise a number of open problems relating to possible differences between the structure of incidences between points and lines versus the structure of incidences between points and pseudolines.

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