Convex-Arc Drawings of Pseudolines

A weak pseudoline arrangement is a topological generalization of a line arrangement, consisting of curves topologically equivalent to lines that cross each other at most once. We consider arrangements that are outerplanar---each crossing is incident to an unbounded face---and simple---each crossing point is the crossing of only two curves. We show that these arrangements can be represented by chords of a circle, by convex polygonal chains with only two bends, or by hyperbolic lines. Simple but non-outerplanar arrangements (non-weak) can be represented by convex polygonal chains or convex smooth curves of linear complexity.Comment: 11 pages, 8 figures. A preliminary announcement of these results was made as a poster at the 21st International Symposium on Graph Drawing, Bordeaux, France, September 2013, and published in Lecture Notes in Computer Science 8242, Springer, 2013, pp. 522--52

Convex-Arc Drawings of Pseudolines ⋆

Introduction. A pseudoline is formed from a line by stretching the plane without tearing: it is the image of a line under a homeomorphism of the plane [13]. In arrangements of pseudolines, pairs of pseudolines intersect at most once and cross at their intersections. Pseudoline arrangements can be used to model sorting networks [1], tilings of convex polygons by rhombi [4], and graphs that have distance-preserving embedding

On the pseudolinear crossing number

A drawing of a graph is {\em pseudolinear} if there is a pseudoline arrangement such that each pseudoline contains exactly one edge of the drawing. The {\em pseudolinear crossing number} of a graph $G$ is the minimum number of pairwise crossings of edges in a pseudolinear drawing of $G$. We establish several facts on the pseudolinear crossing number, including its computational complexity and its relationship to the usual crossing number and to the rectilinear crossing number. This investigation was motivated by open questions and issues raised by Marcus Schaefer in his comprehensive survey of the many variants of the crossing number of a graph.Comment: 12 page

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 $n$ pseudolines we show that the connectivity of the flip graph equals its minimum degree, which is exactly $n-2$. 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 $\Theta(n^3)$. 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

Polyline Drawings with Topological Constraints

Let G be a simple topological graph and let Gamma be a polyline drawing of G. We say that Gamma partially preserves the topology of G if it has the same external boundary, the same rotation system, and the same set of crossings as G. Drawing Gamma fully preserves the topology of G if the planarization of G and the planarization of Gamma have the same planar embedding. We show that if the set of crossing-free edges of G forms a connected spanning subgraph, then G admits a polyline drawing that partially preserves its topology and that has curve complexity at most three (i.e., at most three bends per edge). If, however, the set of crossing-free edges of G is not a connected spanning subgraph, the curve complexity may be Omega(sqrt{n}). Concerning drawings that fully preserve the topology, we show that if G has skewness k, it admits one such drawing with curve complexity at most 2k; for skewness-1 graphs, the curve complexity can be reduced to one, which is a tight bound. We also consider optimal 2-plane graphs and discuss trade-offs between curve complexity and crossing angle resolution of drawings that fully preserve the topology

Geometric Embeddability of Complexes Is ??-Complete

We show that the decision problem of determining whether a given (abstract simplicial) k-complex has a geometric embedding in ?^d is complete for the Existential Theory of the Reals for all d ? 3 and k ? {d-1,d}. Consequently, the problem is polynomial time equivalent to determining whether a polynomial equation system has a real solution and other important problems from various fields related to packing, Nash equilibria, minimum convex covers, the Art Gallery Problem, continuous constraint satisfaction problems, and training neural networks. Moreover, this implies NP-hardness and constitutes the first hardness result for the algorithmic problem of geometric embedding (abstract simplicial) complexes. This complements recent breakthroughs for the computational complexity of piece-wise linear embeddability

Complexity of Some Geometric Problems

We show that recognizing intersection graphs of convex sets has the same complexity as deciding truth in the existential first-order theory of the reals. Comparing this to similar results on the rectilinear crossing number and intersection graphs of line segments, we argue that there is a need to recognize this level of complexity as its own class