Spatial tessellations: concepts and applications of Voronoi diagrams
Spatial tessellations: concepts and applications of Voronoi diagrams
Inequalities for minimal covering sets in set systems of given rank
2nd Twente workshop on Graphs and combinatorial optimization
ICALP '92 Proceedings of the 19th International Colloquium on Automata, Languages and Programming
The class cover problem and its applications in pattern recognition
The class cover problem and its applications in pattern recognition
Computational Statistics & Data Analysis - Data visualization
A new family of proximity graphs: class cover catch digraphs
Discrete Applied Mathematics
Computational Statistics & Data Analysis
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Proximity regions (and maps) are defined based on the relative allocation of points from two or more classes in an area of interest and are used to construct random graphs called proximity catch digraphs (PCDs) which have applications in various fields. The simplest of such maps is the spherical proximity map which gave rise to class cover catch digraph (CCCD) and was applied to pattern classification. In this article, we note some appealing properties of the spherical proximity map in compact intervals on the real line, thereby introduce the mechanism and guidelines for defining new proximity maps in higher dimensions. For non-spherical PCDs, Delaunay tessellation (triangulation in the real plane) is used to partition the region of interest in higher dimensions. We also introduce the auxiliary tools used for the construction of the new proximity maps, as well as some related concepts that will be used in the investigation and comparison of these maps and the resulting PCDs. We provide the distribution of graph invariants, namely, domination number and relative density, of the PCDs and characterize the geometry invariance of the distribution of these graph invariants for uniform data and provide some newly defined proximity maps in higher dimensions as illustrative examples.