12 research outputs found
On the Edge-length Ratio of Outerplanar Graphs
International audienceWe show that any outerplanar graph admits a planar straight-line drawing such that the length ratio of the longest to the shortest edges is strictly less than 2. This result is tight in the sense that for any ε > 0 there are outerplanar graphs that cannot be drawn with an edge-length ratio smaller than 2 −ε. We also show that this ratio cannot be bounded if the embeddings of the outerplanar graphs are given
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 -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 -gon. In this paper we show that the Horton set of
points can be realized with integer coordinates of absolute value at most
. 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 , where is a positive constant
Almost All Even Yao-Yao Graphs Are Spanners
It is an open problem whether Yao-Yao graphs 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 >= 42, the Yao-Yao graph YY_{2k} is a t_k-spanner, with stretch factor t_k = 6.03+O(k^{-1}) when k tends to infinity. Our result generalizes the best known result which asserts that all YY_{6k} are spanners for k >= 6 [Bauer and Damian, SODA\u2713]. Our proof is also somewhat simpler
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem