Related papers: Spanning Properties of Theta-Theta Graphs
We present improved upper bounds on the spanning ratio of constrained $\theta$-graphs with at least 6 cones and constrained Yao-graphs with 5 or at least 7 cones. Given a set of points in the plane, a Yao-graph partitions the plane around…
For a set of points in the plane and a fixed integer $k > 0$, the Yao graph $Y_k$ partitions the space around each point into $k$ equiangular cones of angle $\theta=2\pi/k$, and connects each point to a nearest neighbor in each cone. It is…
We present improved upper and lower bounds on the spanning ratio of $\theta$-graphs with at least six cones. Given a set of points in the plane, a $\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having…
We show that, unlike the Yao-Yao graph $YY_6$, the Theta-Theta graph $\Theta\Theta_6$ defined by six cones is a spanner for sets of points in convex position. We also show that, for sets of points in non-convex position, the spanning ratio…
In this paper, we introduce a variation of the well-studied Yao graphs. Given a set of points $S\subset \mathbb{R}^2$ and an angle $0 < \theta \leq 2\pi$, we define the continuous Yao graph $cY(\theta)$ with vertex set $S$ and angle…
Given a finite set $P\subset\mathbb{R}^2$, the directed Theta-6 graph, denoted $\vec{\Theta}_6(P)$, is a well-studied geometric graph due to its close relationship with the Delaunay triangulation. The $\vec{\Theta}_6(P)$-graph is defined as…
We present tight bounds on the spanning ratio of a large family of ordered $\theta$-graphs. A $\theta$-graph partitions the plane around each vertex into $m$ disjoint cones, each having aperture $\theta = 2 \pi/m$. An ordered $\theta$-graph…
Given a set of points in the plane, we show that the $\theta$-graph with 5 cones is a geometric spanner with spanning ratio at most $\sqrt{50 + 22 \sqrt{5}} \approx 9.960$. This is the first constant upper bound on the spanning ratio of…
We present a routing algorithm for the directed $\Theta_4$-graph, here denoted as the $\overrightarrow{\Theta_4}}$-graph, that computes a path between any two vertices $s$ and $t$ having length at most $17$ times the Euclidean distance…
We show that, for any integer k > 5, the Sparse-Yao graph YY_{6k} (also known as Yao-Yao) is a spanner with stretch factor 11.67. The stretch factor drops down to 4.75 for k > 7.
In this paper we prove that $Y_5$, the Yao graph with five cones, is a spanner with stretch factor $\rho = 2+\sqrt{3} \approx 3.74$. Since $Y_5$ is the only Yao graph whose status of being a spanner or not was open, this completes the…
It is an open problem whether Yao-Yao graphs $\mathsf{YY}_k$ (also known as sparse-Yao graphs) are all spanners when the integer parameter $k$ is large enough. In this paper we show that, for any integer $k\geq 42$, the Yao-Yao graph…
We show an upper bound of $\frac{ \sin\left(\frac{3\pi}{10}\right) }{ \sin\left(\frac{2\pi}{5}\right)-\sin\left(\frac{3\pi}{10}\right) } <5.70$ on the spanning ratio of $\Theta_5$-graphs, improving on the previous best known upper bound of…
We study the problem of embedding graphs in the plane as good geometric spanners. That is, for a graph $G$, the goal is to construct a straight-line drawing $\Gamma$ of $G$ in the plane such that, for any two vertices $u$ and $v$ of $G$,…
Yao graphs are geometric spanners that connect each point of a given point set to its nearest neighbor in each of $k$ cones drawn around it. Yao graphs were introduced to construct minimum spanning trees in $d$ dimensional spaces. Moreover,…
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…
An emanation graph of grade $k$ on a set of points is a plane spanner made by shooting $2^{k+1}$ equally spaced rays from each point, where the shorter rays stop the longer ones upon collision. The collision points are the Steiner points of…
An identifying code of a graph is a dominating set which uniquely determines all the vertices by their neighborhood within the code. Whereas graphs with large minimum degree have small domination number, this is not the case for the…
A Delaunay graph built on a planar point set has an edge between two vertices when there exists a disk with the two vertices on its boundary and no vertices in its interior. When the disk is replaced with an equilateral triangle, the…
Given a directed graph G = (V,E) and an integer k>=1, a k-transitive-closure-spanner (k-TC-spanner) of G is a directed graph H = (V, E_H) that has (1) the same transitive-closure as G and (2) diameter at most k. These spanners were…