Beta-skeletons have unbounded dilation

Beta-skeletons have unbounded dilation

A fractal construction shows that, for any β > 0, the β-skeleton of a point set can have arbitrarily large dilation. In particular this applies to the Gabriel graph.

Generalized $\beta$-skeletons

β-skeletons, a prominent member of the neighborhood graph family, have interesting geometric properties and various applications ranging from geographic networks to archeology. This paper focuses on developing a new, more general than the present one, definition of β-skeletons based only on the distance criterion. It allows us to consider them in many different cases, e.g. for weighted graphs o...

How beta-skeletons lose their edges

A β-skeleton is a proximity graphs with node neighbourhood defined by continuousvalued parameter β. Two nodes in a β-skeleton are connected by an edge if their lune-based neighbourhood contains no other nodes. With increase of β some edges a skeleton are disappear. We study how a number of edges in β-skeleton depends on β. We speculate how this dependence can be used to discriminate between ran...

Directed VR-Representable Graphs have Unbounded Dimension

Visibility representations of graphs map vertices to sets in Euclidean space and express edges as visibility relations between these sets. A three-dimensional visibility representation that has been studied is one in which each vertex of the graph maps to a closed rectangle in R 3 and edges are expressed by vertical visibility between rectangles. The rectangles representing vertices are disjoin...

Directed rectangle visibility graphs have unbounded dimension