Related papers: Finding the theta-Guarded Region

In this paper we show that the \theta-graph with 4 cones has constant stretch factor, i.e., there is a path between any pair of vertices in this graph whose length is at most a constant times the Euclidean distance between that pair of…

We characterize the largest point sets in the plane which define at most 1, 2, and 3 angles. For $P(k)$ the largest size of a point set admitting at most $k$ angles, we prove $P(2)=5$ and $P(3)=5$. We also provide the general bounds of $k+2…

A strong orientation of a graph $G$ is an assignment of a direction to each edge such that $G$ is strongly connected. The oriented diameter of $G$ is the smallest diameter among all strong orientations of $G$. A block of $G$ is a maximal…

Motivated by the work of Douady, Ghys, Herman and Shishikura on Siegel quadratic polynomials, we study the one-dimensional slice of the cubic polynomials which have a fixed Siegel disk of rotation number theta, with theta being a given…

In this paper, we propose a class of elementary plane geometry problems closely related to the title of this paper. Here, a circle is the 1-dimensional curve bounding a disk. For any nonnegative integer, a circle is called $n$-enclosing if…

We prove that for every $\epsilon>0$ there exists $\delta>0$ such that the following holds. Let $\mathcal{C}$ be a collection of $n$ curves in the plane such that there are at most $(\frac{1}{4}-\epsilon)\frac{n^{2}}{2}$ pairs of curves…

Neumann-Lara and Urrutia showed in 1985 that in any set of n points in the plane in general positionthere is always a pair of points such that any circle through them contains at least (n-2)/60 points. In a series of papers, this result was…

Let $\Theta=(\theta_{jk})_{n\times n}$ be a real skew-symmetric $n\times n$ matrix for $n\geq 2$. Under some mild non-integrality conditions on $\Theta,$ we construct Rieffel-type projections as higher dimensional Bott classes in the…

An obstacle representation of a plane graph G is V(G) together with a set of opaque polygonal obstacles such that G is the visibility graph on V(G) determined by the obstacles. We investigate the problem of computing an obstacle…

In the packing-constrained point covering problem, PC^2, one seeks configurations of points in the plane that cannot all be covered by a packing arrangement of unit disks. We consider in particular the problem of finding the minimum number…

The cover time of a finite connected graph is the expected number of steps needed for a simple random walk on the graph to visit all vertices of the graph. It is known that the cover time of any finite connected $n$-vertex graph is at least…

We study the time complexity of the discrete $k$-center problem and related (exact) geometric set cover problems when $k$ or the size of the cover is small. We obtain a plethora of new results: - We give the first subquadratic algorithm for…

Let P be a polygon with rational vertices in the plane. We show that for any finite odd-sized collection of translates of P, the area of the set of points lying in an odd number of these translates is bounded away from 0 by a constant…

Regions in the Euclidean plane surrounded by circles are fundamental geometric and combinatorial objects. Related studies have been done and we cannot explain them precisely, or roughly, well. We study such regions whose Poincar\'e-Reeb…

Let $\Omega \subset \mathbb{R}^2$ be a bounded, convex set. A set $O \subset \mathbb{R}^2$ is an opaque set (for $\Omega$) if every line that intersects $\Omega$ also intersects $O$. What is the minimal possible length $L$ of an opaque set?…

Given a non-negative $n \times n$ matrix viewed as a set of distances between $n$ points, we consider the property testing problem of deciding if it is a metric. We also consider the same problem for two special classes of metrics, tree…

An almost self-centered graph is a connected graph of order $n$ with exactly $n-2$ central vertices, and an almost peripheral graph is a connected graph of order $n$ with exactly $n-1$ peripheral vertices. We determine (1) the maximum girth…

A hole in a graph is a chordless cycle of length at least 4. A theta is a graph formed by three internally vertex-disjoint paths of length at least 2 between the same pair of distinct vertices. A wheel is a graph formed by a hole and a node…

We show that for any $d\ge 2$ and $\Delta>0$ there exists $\eta>0$ such that the following holds: Let $G$ be an $n$-vertex graph with at least $\Omega(n^2)$ edges and let $H$ be an $n$-vertex $d$-degenerate graph with maximum degree at most…

The collision problem is to decide whether a function X:{1,..,n}->{1,..,n} is one-to-one or two-to-one, given that one of these is the case. We show a lower bound of Theta(n^{1/5}) on the number of queries needed by a quantum computer to…