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Article
Open AccessA Tight Bound for the Number of Edges of Matchstick Graphs
A matchstick graph is a plane graph with edges drawn as unit-distance line segments. Harborth introduced these graphs in 1981 and conjectured that the maximum number of edges for a matchstick graph on n vertices ...
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Article
Open AccessAngular measures and Birkhoff orthogonality in Minkowski planes
Let x and y be two unit vectors in a normed plane \(\mathbb {R}^2\) R 2 . We say that x is Birkhoff orthogonal to y if the line through x in the direction y supports the unit disc. A B-measure (Fankhänel i...
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Article
Open AccessAlmost-Equidistant Sets
For a positive integer d, a set of points in d-dimensional Euclidean space is called almost-equidistant if for any three points from the set, some two are at unit distance. Let f(d) denote the largest size of an ...
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Chapter and Conference Paper
Sphere-of-Influence Graphs in Normed Spaces
We show that any k-th closed sphere-of-influence graph in a d-dimensional normed space has a vertex of degree less than \(5^d k\) ...
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Article
Open AccessOn Sets Defining Few Ordinary Circles
An ordinary circle of a set P of n points in the plane is defined as a circle that contains exactly three points of P. We show that if P is not contained in a line or a circle, then P spans at least ...
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Article
Open AccessApproximate Euclidean Steiner Trees
An approximate Steiner tree is a Steiner tree on a given set of terminals in Euclidean space such that the angles at the Steiner points are within a specified error from
