Points In Convex Position Research Articles

Geometric Achromatic and Pseudoachromatic Indices

The pseudoachromatic index of a graph is the maximum number of colors that can be assigned to its edges, such that each pair of different colors is incident to a common vertex. If for each vertex its incident edges have different color, then this maximum is known as achromatic index. Both indices have been widely studied. A geometric graph is a graph drawn in the plane such that its vertices are points in general position, and its edges are straight-line segments. In this paper we extend the notion of pseudoachromatic and achromatic indices for geometric graphs, and present results for complete geometric graphs. In particular, we show that for $n$ points in convex position the achromatic index and the pseudoachromatic index of the complete geometric graph are $\lfloor \tfrac{n^2+n}{4} \rfloor$.

Disjoint Compatibility Graph of Non-Crossing Matchings of Points in Convex Position

Let $X_{2k}$ be a set of $2k$ labeled points in convex position in the plane. We consider geometric non-intersecting straight-line perfect matchings of $X_{2k}$. Two such matchings, $M$ and $M'$, are disjoint compatible if they do not have common edges, and no edge of $M$ crosses an edge of $M'$. Denote by $\rm{DCM}_k$ the graph whose vertices correspond to such matchings, and two vertices are adjacent if and only if the corresponding matchings are disjoint compatible. We show that for each $k \geq 9$, the connected components of $\rm{DCM}_k$ form exactly three isomorphism classes - namely, there is a certain number of isomorphic small components, a certain number of isomorphic medium components, and one big component. The number and the structure of small and medium components is determined precisely.

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

This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of non-crossing spanning trees and forests. We show that the so-called double chain point configuration of N points has Ω(12.52N) non-crossing spanning trees and Ω(13.61N) non-crossing forests. This improves the previous lower bounds on the maximum number of non-crossing spanning trees and of non-crossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Tóth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12N) for the number of non-crossing spanning trees of the double chain is also obtained.

On the number of occurrences of the [formula omitted]th smallest distance between points in convex position

Let P be a set of n points in the plane in convex position and let d1<d2<⋯ be the distances occurring between two different points of P. Let nk be the number of pairs in P with distance dk. We prove that if n is sufficiently large given k, then nk≤4n.

NOTE ON THE NUMBER OF OBTUSE ANGLES IN POINT SETS

In 1979 Conway, Croft, Erdős and Guy proved that every set S of n points in general position in the plane determines at least [Formula: see text] obtuse angles and also presented a special set of n points to show the upper bound [Formula: see text] on the minimum number of obtuse angles among all sets S. We prove that every set S of n points in convex position determines at least [Formula: see text] obtuse angles, hence matching the upper bound (up to sub-cubic terms) in this case. Also on the other side, for point sets with low rectilinear crossing number, the lower bound on the minimum number of obtuse angles is improved.

Chromatic numbers of copoint graphs of convex geometries

We study the copoint graph of a convex geometry. We give a family of copoint graphs for which the ratio of the chromatic number to the clique number can be arbitrarily large. For any natural numbers 1<d<k, we study the existence of a number Kd(k) so that the chromatic number of the copoint graph of a convex geometry on a set of at least Kd(k) elements, with every d-element subset closed, has chromatic number at least k. Our results are analogues of results of Erdős and Szekeres for convex geometries realizable by point sets in Rd, where cliques in the copoint graph correspond to subsets of points in convex position.

Convex geometric [formula omitted]-quasiplanar representations of semi-bar [formula omitted]-visibility graphs

Semi-bar k-visibility graphs are graphs for which vertices can be drawn as horizontal segments with left endpoints on the y-axis (semi-bars) and edges can be drawn as vertical segments (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. Cyclic semi-bar k-visibility graphs are graphs for which vertices can be drawn as segments with one endpoint at the origin (semi-bars) and edges can be drawn as arcs of circles centered at the origin (sightlines) so that two semi-bars are visible to each other if and only if there is a sightline which intersects them and at most k other semi-bars. We show that every semi-bar or cyclic semi-bar k-visibility graph can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k+2 pairwise crossing edges. Furthermore, we prove that the class of graphs having cyclic semi-bar k-visibility representations with semi-bars of different lengths is the same as the class of (2k+2)-degenerate graphs that can be represented in the plane with vertices drawn as points in convex position and edges drawn as segments so that there are no k+2 pairwise crossing edges and so that adding any edge would result in k+2 pairwise crossing edges.

The Erdős-Sós conjecture for geometric graphs

Combinatorics Let f(n,k) be the minimum number of edges that must be removed from some complete geometric graph G on n points, so that there exists a tree on k vertices that is no longer a planar subgraph of G. In this paper we show that ( 1 / 2 )n2 / k-1-n / 2≤f(n,k) ≤2 n(n-2) / k-2. For the case when k=n, we show that 2 ≤f(n,n) ≤3. For the case when k=n and G is a geometric graph on a set of points in convex position, we completely solve the problem and prove that at least three edges must be removed.

Flip Graphs of Bounded Degree Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.

Erdős–Szekeres Theorem for Point Sets with Forbidden Subconfigurations

According to the Erdős–Szekeres theorem, every set of n points in the plane contains roughly logn points in convex position. We investigate how this bound changes if our point set does not contain a subset that belongs to a fixed order type.

Flipping triangles and rectangles

We study the chromatic number of the flip graph of triangles determined by n points in convex position in the plane, and present new or improved bounds on several related parameters for this graph. We also find the chromatic numbers of two related graphs: the rectangle flip graph which generalizes the shift graph, and the rolling block graph from the popular puzzle "Rolling Block Maze."

Neighbourly polytopes with few vertices

A family of neighbourly polytopes in with vertices is constructed. All polytopes in the family have a planar Gale diagram of a special type, namely, with exactly black points in convex position. These Gale diagrams are parametrized by -trees (trees with a certain additional structure). For all polytopes in the family, the number of faces of dimension containing a given vertex depends only on and .Bibliography: 7 titles.

CONSTRAINED POINT-SET EMBEDDABILITY OF PLANAR GRAPHS

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).

On the stretch factor of Delaunay triangulations of points in convex position

Let S be a finite set of points in the Euclidean plane. Let G be a geometric graph in the plane whose point set is S. The stretch factor of G is the maximum ratio, among all points p and q in S, of the length of the shortest path from p to q in G over the Euclidean distance | pq | . Keil and Gutwin in 1989 [11] proved that the stretch factor of the Delaunay triangulation of a set of points S in the plane is at most 2 π / ( 3 cos ( π / 6 ) ) ≈ 2.42 . Improving on this upper bound remains an intriguing open problem in computational geometry. In this paper we consider the special case when the points in S are in convex position. We prove that in this case the stretch factor of the Delaunay triangulation of S is at most ρ = 2.33 .

Flip Graphs of Bounded-Degree Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant k. Specifically, we consider triangulations of sets of n points in convex position in the plane and prove that their flip graph is connected if and only if k>6; the diameter of the flip graph is O(n2). We also show that for general point sets, flip graphs of triangulations with degree ⩽k can be disconnected for any k.

Extremal problems on triangle areas in two and three dimensions

The study of extremal problems on triangle areas was initiated in a series of papers by Erdős and Purdy in the early 1970s. In this paper we present new results on such problems, concerning the number of triangles of the same area that are spanned by finite point sets in the plane and in 3-space, and the number of distinct areas determined by the triangles. In the plane, our main result is an O ( n 44 / 19 ) = O ( n 2.3158 ) upper bound on the number of unit-area triangles spanned by n points, which is the first breakthrough improving the classical bound of O ( n 7 / 3 ) from 1992. We also make progress in a number of important special cases. We show that: (i) For points in convex position, there exist n-element point sets that span Ω ( n log n ) triangles of unit area. (ii) The number of triangles of minimum (nonzero) area determined by n points is at most 2 3 ( n 2 − n ) ; there exist n-element point sets (for arbitrarily large n) that span ( 6 / π 2 − o ( 1 ) ) n 2 minimum-area triangles. (iii) The number of acute triangles of minimum area determined by n points is O ( n ) ; this is asymptotically tight. (iv) For n points in convex position, the number of triangles of minimum area is O ( n ) ; this is asymptotically tight. (v) If no three points are allowed to be collinear, there are n-element point sets that span Ω ( n log n ) minimum-area triangles (in contrast to (ii), where collinearities are allowed and a quadratic lower bound holds). In 3-space we prove an O ( n 17 / 7 β ( n ) ) = O ( n 2.4286 ) upper bound on the number of unit-area triangles spanned by n points, where β ( n ) is an extremely slowly growing function related to the inverse Ackermann function. The best previous bound, O ( n 8 / 3 ) , is an old result of Erdős and Purdy from 1971. We further show, for point sets in 3-space: (i) The number of minimum nonzero area triangles is at most n 2 + O ( n ) , and this is worst-case optimal, up to a constant factor. (ii) There are n-element point sets that span Ω ( n 4 / 3 ) triangles of maximum area, all incident to a common point. In any n-element point set, the maximum number of maximum-area triangles incident to a common point is O ( n 4 / 3 + ε ) , for any ε > 0 . (iii) Every set of n points, not all on a line, determines at least Ω ( n 2 / 3 / β ( n ) ) triangles of distinct areas, which share a common side.

On the chromatic number of some flip graphs

Graphs and Algorithms This paper studies the chromatic number of the following four flip graphs (under suitable definitions of a flip): the flip graph of perfect matchings of a complete graph of even order, the flip graph of triangulations of a convex polygon (the associahedron), the flip graph of non-crossing Hamiltonian paths of a set of points in convex position, and the flip graph of triangles in a convex point set. We give tight bounds for the latter two cases and upper bounds for the first two.

On the diameter of geometric path graphs of points in convex position

For a set S of n points in convex position in the plane, let P ( S ) denote the set of all plane spanning paths of S. The geometric path graph of S, denoted by G n , is the graph with P ( S ) as its vertex set and two vertices P , Q ∈ P ( S ) are adjacent if and only if P and Q can be transformed to each other by means of a single edge replacement. Recently, Akl et al. [S.G. Akl, K. Islam, H. Meijer, On planar path transformation, Inform. Process. Lett. 104 (2007) 59–64] showed that the diameter of G n is at most 2 n − 5 . In this note, we derive the exact diameter of G n for n ⩾ 3 .

Planar bichromatic minimum spanning trees

Given a set S of n red and blue points in the plane, a planar bichromatic minimum spanning tree is the shortest possible spanning tree of S, such that every edge connects a red and a blue point, and no two edges intersect. We show that computing this tree is NP-hard in general. For points in convex position, a cubic-time algorithm can be easily designed using dynamic programming. We adapt such an algorithm for the special case where the number of red points (m) is much smaller than the number of blue points (n), resulting in an O(nm2) time algorithm. For the general case, we present a factor O(n) approximation algorithm that runs in O(nlognloglogn) time. Finally, we show that if the number of points in one color is bounded by a constant, the optimal tree can be computed in polynomial time.

Erdős–Szekeres “happy end”-type theorems for separoïds

In 1935 Pál Erdős and György Szekeres proved that, roughly speaking, any configuration of n points in general position in the plane have log n points in convex position — which are the vertices of a convex polygon. Later, in 1983, Bernhard Korte and László Lovász generalised this result in a purely combinatorial context; the context of greedoids. In this note we give one step further to generalise this last result for arbitrary dimensions, but in the context of separoids; thus, via the geometric representation theorem for separoids, this can be applied to families of convex bodies. Also, it is observed that the existence of some homomorphisms of separoids implies the existence of not-too-small polytopal subfamilies — where each body is separated from its relative complement. Finally, by means of a probabilistic argument, it is settled, basically, that for all d > 2 , asymptotically almost all “simple” families of n “ d - separated” convex bodies contains a polytopal subfamily of order log n d + 1 .