Noncrossing Hamiltonian Paths in Geometric Graphs

A geometric graph is a graph drawn in the plane so that the vertices are represented by points in general position, the edges are represented by straight line segments connecting the corresponding points.

The paper bounds the combinatorial complexity of the Voronoi diagram of a set of points under certain polyhedral distance functions. Specifically, if S is a set of n points in general position in R d , the maximum complexity of its Voronoi diagram under the L ∞ metric, and also under a simplicial distance function, are both shown to be \(\Theta(n^{\lceil d/2 \rceil})\) . The upper bound for the case of the L ∞ metric follows from a new upper bound, also proved in this paper, on the maximum complexity of the union of n axis-parallel hypercubes in R d . This complexity is \(\Theta(n^{\left\lceil d/2 \right\rceil})\) , for d ≥ 1 , and it improves to \(\Theta(n^{\left\lfloor d/2 \right\rfloor})\) , for d ≥ 2 , if all the hypercubes have the same size. Under the L 1 metric, the maximum complexity of the Voronoi diagram of a set of n points in general position in R 3 is shown to be \(\Theta(n^2)\) . We also show that the general position assumption is essential, and give examples where the complexity of the diagram increases significantly when the points are in degenerate configurations. (This increase does not occur with an appropriate modification of the diagram definition.) Finally, on-line algorithms are proposed for computing the Voronoi diagram of n points in Rd under a simplicial or L ∞ distance function. Their expected randomized complexities are \(O(n \log n + n ^{\left\lceil d/2 \right\rceil})\) for simplicial diagrams and \(O(n ^{\left\lceil d/2 \right\rceil} \log ^{d-1} n)\) for L ∞ -diagrams.

Lower bounds on the maximum number of non-crossing acyclic graphs

On embedding triconnected cubic graphs on point sets

A geometric graph is a simple graph G=(V,E) with an embedding of the set V in the plane such that the points that represent V are in general position. A geometric graph is said to be k-locally Delaunay (or a Dk-graph) if for each edge (u,v) ∈ E there is a (Euclidean) disc d that contains u and v but no other vertex of G that is within k hops from u or v.The study of these graphs was recently motivated by topology control for wireless networks [6,7]. We obtain the following results: (i) We prove that if G is a D1-graph on n vertices, then it has O(n3/2) edges. (ii) We show that for any n there exist D1-graphs with n vertices and Ω(n4/3) edges. (iii) We prove that if G is a D2-graph on n vertices, then it has O(n) edges. This bound is worst-case asymptotically tight. As an application of the first result, we show that: (iv) The maximum size of a family of pairwise non-overlapping lenses in an arrangement of $n$ unit circles in the plane is O(n3/2).The first two results improve the best previously known upper and lower bounds of $O(n^ 5/3 )$ and $\Omega(n)$ respectively (see \cite KL03 ). The third result improves the best previously known upper bound of O(n log n ) ([6]). Finally, our last result improves the best previously known upper bound (for the more general case of not necessarily unit circles) of O(n3/2 κ(n)) (see [1] ), where κ(n) = (log n ) O(α2(n)) and where α(n) is the extremely slowly growing inverse Ackermann's function.

The classical Cover results on linear separability of points in Rd are a milestone in neural network theory. Nevertheless they are not valid for digital input networks because in this case the points are not, in general position. vertices ofa d-dimensional hypercube. I show here that for large d all Cover findings can be extended to this case. It is also shown that for n < O((d + I)*) the number of linear separations of n random hypercube vertices tends to that of n points in general position.

This paper starts the investigation of a constrained version of the point-set embed-dability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: • If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E\E′ have at most 4 bends each. • If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E\E′ is at most 8. • If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E \ E′ is O(2k).

Hamiltonian cycles and paths in vertex-transitive graphs with abelian and nilpotent groups

Order types are a means to characterize the combinatorial properties of a finite point configuration. In particular, the crossing properties of all straight-line segments spanned by an planar $n$-point set are reflected by its order type. We establish a complete and reliable data base for all possible order types of size $n=10$ or less. The data base includes a realizing point set for each order type in small integer grid representation. To our knowledge, no such project has been carried out before.We substantiate the usefulness of our data base by applying it to several problems in computational and combinatorial geometry. Problems concerning triangulations, simple polygonalizations, complete geometric graphs, and $k$-sets are addressed. This list of possible applications is not meant to be exhaustive. We believe our data base to be of value to many researchers who wish to examine their conjectures on small point configurations.

Kontsevich's formula is a recursion that calculates the number of rational degree $d$ curves in $\mathbb{P}_{\mathbb{C}}^2$ passing through $3d-1$ points in general position. Kontsevich proved it by considering curves that satisfy extra conditions besides the given point conditions. These crucial extra conditions are two line conditions and a condition called cross-ratio.&#x0D; This paper addresses the question whether there is a general Kontsevich's formula which holds for more than one cross-ratio. Using tropical geometry, we obtain such a recursive formula. For that, we use a correspondence theorem of Tyomkin that relates the algebro-geometric numbers in question to tropical ones. It turns out that the general tropical Kontsevich's formula we obtain is capable of not only computing the algebro-geometric numbers we are looking for, but also of computing further tropical numbers for which there is no correspondence theorem yet.&#x0D; We show that our recursive general Kontsevich's formula implies the original Kontsevich's formula and that the initial values are the numbers Kontsevich's fomula provides and purely combinatorial numbers, so-called cross-ratio multiplicities.

A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (−1) curve is an irreducible curve with self intersection −1 and genus zero. The Segre–Harbourne–Hirschowitz Conjecture states that if a linear system is special then a multiple of some fixed (−1) curve is contained in every curve of the linear system. This conjecture is proven for linear systems with multiplicity four at all but one of the points.

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.

New cases of Reay’s conjecture on partitions of points into simplices with [formula omitted]-dimensional intersection

Let $\Pi$ be a convex decomposition of a set $P$ of $n\geq 3$ points in general position in the plane. If $\Pi$ consists of more than one polygon, then either $\Pi$ contains a deletable edge or $\Pi$ contains a contractible edge.

Resolutions of small sets of fat points