Properties of some Euclidean proximity graphs
Pattern Recognition Letters
Classes of graphs which approximate the complete Euclidean graph
Discrete & Computational Geometry
Relative neighborhood graphs in three dimensions
Computational Geometry: Theory and Applications
Computational Geometry: Theory and Applications
Proximity Drawability: a Survey
GD '94 Proceedings of the DIMACS International Workshop on Graph Drawing
On the Spanning Ratio of Gabriel Graphs and beta-Skeletons
SIAM Journal on Discrete Mathematics
Proceedings of the nineteenth annual ACM-SIAM symposium on Discrete algorithms
On the Inequality of Cover and Hart in Nearest Neighbor Discrimination
IEEE Transactions on Pattern Analysis and Machine Intelligence
On a family of strong geometric spanners that admit local routing strategies
Computational Geometry: Theory and Applications
Locally-scaled spectral clustering using empty region graphs
Proceedings of the 18th ACM SIGKDD international conference on Knowledge discovery and data mining
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A family of proximity graphs, called Empty Region Graphs (ERG) is presented. The vertices of an ERG are points in the plane, and two points are connected if their neighborhood, defined by a region, does not contain any other point. The region defining the neighborhood of two points is a parameter of the graph. This way of defining graphs is not new, and ERGs include several known proximity graphs such as Nearest Neighbor Graphs, @b-Skeletons or @Q-Graphs. The main contribution is to provide insight and connections between the definition of ERG and the properties of the corresponding graphs. We give conditions on the region defining an ERG to ensure a number of properties that might be desirable in applications, such as planarity, connectivity, triangle-freeness, cycle-freeness, bipartiteness and bounded degree. These conditions take the form of what we call tight regions: maximal or minimal regions that a region must contain or be contained in to make the graph satisfy a given property. We show that every monotone property has at least one corresponding tight region; we discuss possibilities and limitations of this general model for constructing a graph from a point set.