104 research outputs found
Odd Yao-Yao Graphs are Not Spanners
It is a long standing open problem whether Yao-Yao graphs YY_{k} are all spanners [Li et al. 2002]. Bauer and Damian [Bauer and Damian, 2012] showed that all YY_{6k} for k >= 6 are spanners. Li and Zhan [Li and Zhan, 2016] generalized their result and proved that all even Yao-Yao graphs YY_{2k} are spanners (for k >= 42). However, their technique cannot be extended to odd Yao-Yao graphs, and whether they are spanners are still elusive. In this paper, we show that, surprisingly, for any integer k >= 1, there exist odd Yao-Yao graph YY_{2k+1} instances, which are not spanners
Continuous Yao Graphs
In this paper, we introduce a variation of the well-studied Yao graphs. Given
a set of points and an angle , we
define the continuous Yao graph with vertex set and angle
as follows. For each , we add an edge from to in
if there exists a cone with apex and aperture such
that is the closest point to inside this cone.
We study the spanning ratio of for different values of .
Using a new algebraic technique, we show that is a spanner when
. We believe that this technique may be of independent
interest. We also show that is not a spanner, and that
may be disconnected for .Comment: 7 pages, 7 figures. Presented at CCCG 201
Theta-3 is connected
In this paper, we show that the -graph with three cones is connected.
We also provide an alternative proof of the connectivity of the Yao graph with
three cones.Comment: 11 pages, to appear in CGT
There are Plane Spanners of Maximum Degree 4
Let E be the complete Euclidean graph on a set of points embedded in the
plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a
t-spanner, or simply a spanner, if for any pair of vertices u,v in E the
distance between u and v in G is at most t times their distance in E. A spanner
is plane if its edges do not cross.
This paper considers the question: "What is the smallest maximum degree that
can always be achieved for a plane spanner of E?" Without the planarity
constraint, it is known that the answer is 3 which is thus the best known lower
bound on the degree of any plane spanner. With the planarity requirement, the
best known upper bound on the maximum degree is 6, the last in a long sequence
of results improving the upper bound. In this paper we show that the complete
Euclidean graph always contains a plane spanner of maximum degree at most 4 and
make a big step toward closing the question. Our construction leads to an
efficient algorithm for obtaining the spanner from Chew's L1-Delaunay
triangulation
Spanning Properties of Theta-Theta-6
We show that, unlike the Yao–Yao graph YY6, the Theta–Theta graph ΘΘ6 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 of ΘΘ6 is unbounded
Spanning Properties of Theta-Theta-6
We show that, unlike the Yao-Yao graph , the Theta-Theta graph
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 of is unbounded.Comment: 13 pages, 9 figure
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