An upper bound for conforming Delaunay triangulations

A plane geometric graphC in ℝ² conforms to another such graphG if each edge ofG is the union of some edges ofC. It is proved that, for everyG withn vertices andm edges, there is a completion of a Delaunay triangulation ofO(m ² n) points that conforms toG. The algorithm that constructs the points is also described. Research of the first author is supported by the National Science Foundation under Grant CCR-8921421 and under the Alan T. Waterman award, Grant CCR-9118874. Any opinions, findings, and conclusions or recommendations expressed in this publication are those of the authors and do not necessarily reflect the view of the National Science Foundation. Work of the second author was conducted while he was on study leave at the University of Illinois.