Drawing the Horton Set in an Integer Grid of Minimum Size

In 1978 Erd\H os asked if every sufficiently large set of points in general position in the plane contains the vertices of a convex $k$-gon, with the additional property that no other point of the set lies in its interior. Shortly after, Horton provided a construction---which is now called the Horton set---with no such $7$-gon. In this paper we show that the Horton set of $n$ points can be realized with integer coordinates of absolute value at most $\frac{1}{2} n^{\frac{1}{2} \log (n/2)}$. We also show that any set of points with integer coordinates combinatorially equivalent (with the same order type) to the Horton set, contains a point with a coordinate of absolute value at least $c \cdot n^{\frac{1}{24}\log (n/2)}$, where $c$ is a positive constant

On kk-Gons and kk-Holes in Point Sets

We consider a variation of the classical Erd\H{o}s-Szekeres problems on the existence and number of convex $k$-gons and $k$-holes (empty $k$-gons) in a set of $n$ points in the plane. Allowing the $k$-gons to be non-convex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any $k$ and sufficiently large $n$, we give a quadratic lower bound for the number of $k$-holes, and show that this number is maximized by sets in convex position

Sparse geometric graphs with small dilation

Given a set S of n points in R^D, and an integer k such that 0 <= k < n, we show that a geometric graph with vertex set S, at most n - 1 + k edges, maximum degree five, and dilation O(n / (k+1)) can be computed in time O(n log n). For any k, we also construct planar n-point sets for which any geometric graph with n-1+k edges has dilation Omega(n/(k+1)); a slightly weaker statement holds if the points of S are required to be in convex position

Data structures for retrieval on integer grids

A family of data structures is presented for retrieval of the sum of values of points within a half-plane or polygon, given that the points are on integer coordinates in the plane. Fredman has shown that the problem has a lower bound of Ω(N^2/3) for intermixed updates and retrievals. Willard has shown an upper bound of O(N^2log6^4) for the case where the points are not restricted to integer coordinates.We have developed families of related data structures for retrievals of half-planes or polygons. One of the data structures permits intermixed updates and half-plane retrievals in O(N^2/3log N) time, where N is the size of the grid.We use a technique we call "Rotation" to permit a better match of a portion of the data structure to the particular problem. Rotations appear to be an effective method for trading-off storage redundancy against retrieval time for certain classes of problems

Stabbing line segments with disks: complexity and approximation algorithms

Computational complexity and approximation algorithms are reported for a problem of stabbing a set of straight line segments with the least cardinality set of disks of fixed radii $r>0$ where the set of segments forms a straight line drawing $G=(V,E)$ of a planar graph without edge crossings. Close geometric problems arise in network security applications. We give strong NP-hardness of the problem for edge sets of Delaunay triangulations, Gabriel graphs and other subgraphs (which are often used in network design) for $r\in [d_{\min},\eta d_{\max}]$ and some constant $\eta$ where $d_{\max}$ and $d_{\min}$ are Euclidean lengths of the longest and shortest graph edges respectively. Fast $O(|E|\log|E|)$-time $O(1)$-approximation algorithm is proposed within the class of straight line drawings of planar graphs for which the inequality $r\geq \eta d_{\max}$ holds uniformly for some constant $\eta>0,$ i.e. when lengths of edges of $G$ are uniformly bounded from above by some linear function of $r.$Comment: 12 pages, 1 appendix, 15 bibliography items, 6th International Conference on Analysis of Images, Social Networks and Texts (AIST-2017

Combinatorics and geometry of finite and infinite squaregraphs

Squaregraphs were originally defined as finite plane graphs in which all inner faces are quadrilaterals (i.e., 4-cycles) and all inner vertices (i.e., the vertices not incident with the outer face) have degrees larger than three. The planar dual of a finite squaregraph is determined by a triangle-free chord diagram of the unit disk, which could alternatively be viewed as a triangle-free line arrangement in the hyperbolic plane. This representation carries over to infinite plane graphs with finite vertex degrees in which the balls are finite squaregraphs. Algebraically, finite squaregraphs are median graphs for which the duals are finite circular split systems. Hence squaregraphs are at the crosspoint of two dualities, an algebraic and a geometric one, and thus lend themselves to several combinatorial interpretations and structural characterizations. With these and the 5-colorability theorem for circle graphs at hand, we prove that every squaregraph can be isometrically embedded into the Cartesian product of five trees. This embedding result can also be extended to the infinite case without reference to an embedding in the plane and without any cardinality restriction when formulated for median graphs free of cubes and further finite obstructions. Further, we exhibit a class of squaregraphs that can be embedded into the product of three trees and we characterize those squaregraphs that are embeddable into the product of just two trees. Finally, finite squaregraphs enjoy a number of algorithmic features that do not extend to arbitrary median graphs. For instance, we show that median-generating sets of finite squaregraphs can be computed in polynomial time, whereas, not unexpectedly, the corresponding problem for median graphs turns out to be NP-hard.Comment: 46 pages, 14 figure