Richard Pollack

How to draw a planar graph on a grid

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph withn vertices has a Fáry embedding (i.e., straight-line embedding) on the 2n−4 byn−2 grid and provide anO(n) space,O(n logn) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any setF, which can support a Fáry embedding of every planar graph of sizen, has cardinality at leastn+(1−o(1))√n which settles a problem of Mohar.

Small sets supporting fary embeddings of planar graphs

Answering a question of Rosenstiehl and Tarjan, we show that every plane graph with <italic>n</italic> vertices has a Fáry embedding (i.e., straight-line embedding) on the 2<italic>n</italic> - 4 by <italic>n</italic> - 2 grid and provide an &Ogr;(<italic>n</italic>) space, &Ogr;(<italic>n</italic> log <italic>n</italic>) time algorithm to effect this embedding. The grid size is asymptotically optimal and it had been previously unknown whether one can always find a polynomial sized grid to support such an embedding. On the other hand we show that any set <italic>F</italic>, which can support a Fáry embedding of every planar graph of size <italic>n</italic>, has cardinality at least <italic>n</italic> + (1 - <italic>&ogr;</italic>(1)) √<italic>n</italic> which settles a problem of Mohar.

On the combinatorial classification of nondegenerate configurations in the plane

Abstract We classify nondegenerate plane configurations by attaching, to each such configuration of n points, a periodic sequence of permutations of {1, 2, …, n } which satisfies some simple conditions; this classification turns out to be appropriate for questions involving convexity. In 1881 Perrin stated that every sequence satisfying these conditions was the image of some plane configuration. We show that this statement is incorrect by exhibiting a counterexample, for n = 5, and prove that for n ⩽ 5 every sequence essentially distinct from this one is realized geometrically by giving a complete classification of configurations in these cases; there is 1 combinatorial equivalence class for n = 3, 2 for n = 4, and 19 for n = 5. We develop some basic notions of the geometry of “allowable sequences” in the course of proving this classification theorem. Finally, we state some results and an open problem on the realizability question in the general case.

Allowable Sequences and Order Types in Discrete and Computational Geometry

The allowable sequence associated to a configuration of points was first developed by the authors in order to investigate what combinatorial structure lay behind the Erdős-Szekeres conjecture (that any 2 n-2 + 1 points in general position in the plane contain among them n points which are in convex position). Though allowable sequences did not lead to any progress on this ancient problem, there did emerge an object that had considerable intrinsic interest, that turned out to be related to some other well-studied structures such as pseudoline arrangements and oriented matroids, and that had as well a combinatorial simplicity and suggestiveness which turned out to be effective in the solution of several other classical problems. These connections and applications are discussed in Sections 2, 3, and 4 of this paper.

Upper bounds for configurations and polytopes inRd

We give a new upper bound onnd(d+1)n on the number of realizable order types of simple configurations ofn points inRd, and ofn2d2n on the number of realizable combinatorial types of simple configurations. It follows as a corollary of the first result that there are no more thannd(d+1)n combinatorially distinct labeled simplicial polytopes inRd withn vertices, which improves the best previous upper bound ofncnd/2.

Computing roadmaps of semi-algebraic sets on a variety

Let R be a real closed field, Z(Q) a real algebraic variety defined as the zero set of a polynomial Q ∈ R[X1, . . . , Xk] and S a semi-algebraic subset of Z(Q), defined by a Boolean formula with atoms of the form P < 0, P > 0, P = 0 with P ∈ P , where P is a finite subset of R[X1, . . . , Xk]. A semi-algebraic set C is semi-algebraically connected if it is non-empty and is not the union of two non-empty disjoint semi-algebraic sets which are closed and open in C. A semi-algebraically connected component of S is a semi-algebraic subset of S which is semi-algebraically connected, and closed and open in S . Semi-algebraic sets have a finite number of semi-algebraically connected components ([5], page 34). A roadmap of S, which we denote R(S), is a semi-algebraic set of dimension at most one contained in S which satisfies the roadmap conditions: RM1 For every semi-algebraically connected component C of S, C ∩R(S) is semialgebraically connected. RM2 For every x ∈ R, and for every semi-algebraically connected component C′ of Sx, C ′ ∩R(S) 6= ∅. Here, and everywhere else in this paper, π is the projection on the first coordinate and for X ⊂ R, SX is S ∩ π−1(X). We also use the abbreviations Sx, S<c, and S≤c for S{x}, S(−∞,c), and S(−∞,c] respectively. Algorithms for the construction of roadmaps are described in terms of the parameters k, k′, s, d where k is the dimension of the ambient space, k′ is the dimension of Z(Q), s is the number of polynomials in P and d is a bound on the degrees of the polynomials in P and the polynomial Q. Given a roadmap R(S) and a point p ∈ S the connecting subroutine outputs a semi-algebraic continuous path in Sπ(p) connecting p to R(S). The connecting subroutine is described in terms of the parameters k, k′, s, d as before and τ which is a bound on the degrees of the polynomials defining p (see section 4 for a discussion of how points are described by polynomials).

Computing the geodesic center of a simple polygon

The geodesic center of a simple polygon is a point inside the polygon which minimizes the maximum internal distance to any point in the polygon. We present an algorithm which calculates the geodesic center of a simple polygon withn vertices in timeO(n logn).

Proof of Grünbaum's conjecture on the stretchability of certain arrangements of pseudolines

Abstract We prove Grunbaums conjecture that every arrangement of eight pseudolines in the projective plane is stretchable, i.e., determines a cell complex isomorphic to one determined by an arrangement of lines. The proof uses our previous results on ordered duality in the projective plane and on periodic sequences of permutations of [1,n] associated to arrangements of n lines in the euclidean plane.

Coordinate representation of order types requires exponential storage

We give doubly exponential upper and lower bounds on the size of the smallest grid on which we can embed every planar configuration of n points in general position up to order type. The lower bound is achieved by the construction of a widely dispersed “rigid” configuration which is then modified to one in general position by recent techniques of Sturmfels and White, while the upper bound uses recent results of Grigorev and Vorobjou on the solution of simultaneous inequalities. This provides a sharp answer to a question first posed by Chazelle.

Helly-Type Theorems for Pseudoline Arrangements in P2

Abstract We prove duals of Radons theorem, Hellys theorem, Caratheodorys theorem, and Kirchbergers theorem for arrangements of pseudolines in the real projective plane, which generalize the original versions of those theorems for plane configurations of points. We also prove a topological generalization of the pseudoline-dual of Hellys theorem.