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We consider two types of geometric graphs on point sets on the plane based ona plane set C: one obtained by translates of C, another by positively scaledtranslates (homothets) of C. For compact and convex C, graphs defined by scaledtranslates of C, i.e., Delaunay graphs based on C, are known to be planegraphs. We show that as long as C is convex, both types of graphs are planegraphs.

Generalized Delaunay Graphs with respect to any Convex Set are Plane Graphs