The relative neighbourhood graph is a part of every 30°-triangulation

Authors:
J. Mark Keil;Tzvetalin S. Vassilev
Affiliations:
Department of Computer Science, University of Saskatchewan, 110 Science Place, Saskatoon, Saskatchewan, S7N 5C9, Canada;Department of Mathematics and Computer Science, North Carolina Central University, 1801 Fayetteville Street, Durham, NC 27707, USA
Venue:
Information Processing Letters
Year:
2008

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Abstract

We study sets of points in the two-dimensional Euclidean plane. The relative neighbourhood graph (RNG) of a point set is a straight line graph that connects two points from the point set if and only if there is no other point in the set that is closer to both points than they are to each other. A triangulation of a point set is a maximal set of nonintersecting line segments (called edges) with vertices in the point set. We introduce angular restrictions in the triangulations. Using the well-known method of exclusion regions, we show that the relative neighbourhood graph is a part of every triangulation all of the angles of which are greater than or equal to 30^o.