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

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

💡 Research Summary

The paper investigates two families of geometric graphs that are defined on a finite set of points P in the Euclidean plane with respect to a fixed planar set C. The first family, which we call the C‑translate graph G₁(P,C), connects two points p and q by an edge whenever there exists a translation of C that contains p and q and no other point of P. The second family, the C‑homothet graph G₂(P,C), does the same but allows the copy of C to be uniformly scaled by a positive factor (a homothet) before translation.

For compact convex sets C, it has long been known that G₂ is a planar graph; the proof relies on the empty‑region property of Delaunay graphs and the existence of a C‑Voronoi diagram that is dual to a planar triangulation. What was not known before this work is whether the weaker condition “C is convex” (without compactness) is sufficient for planarity, and whether the translation‑only version G₁ enjoys the same property.

The authors first formalize the two graph definitions and recall the classic empty‑region characterization: an edge (p,q) belongs to the graph if and only if there exists a copy of C that is empty of all other points of P while containing p and q. They then prove two main theorems:

1. Theorem 1 (Planarity of G₁). If C is convex, the C‑translate graph G₁(P,C) is planar. The proof proceeds by contradiction. Assume two edges (p₁,q₁) and (p₂,q₂) cross. Let C₁ and C₂ be the translated copies of C that certify each edge. Their intersection contains the crossing point. Because C is convex, the intersection C₁∩C₂ is also convex and contains a small neighbourhood around the crossing point. This neighbourhood must contain at least three of the four endpoints, contradicting the emptiness condition for at least one of the two copies. Hence no crossing can occur. The argument does not require C to be bounded; it works for open, closed, or unbounded convex sets alike.

2. Theorem 2 (Planarity of G₂ under convexity). If C is convex (compactness is not needed), the C‑homothet graph G₂(P,C) is planar. The authors observe that for any two homothets λ₁C+v₁ and λ₂C+v₂ that certify crossing edges, the monotonicity of scaling forces one homothet to be contained in the other. Consequently the larger homothet would contain three of the four endpoints, violating the empty‑region condition. This yields a contradiction and establishes planarity.

The paper also discusses how these results extend the classical Delaunay theory. While the traditional Delaunay triangulation corresponds to the case where C is a Euclidean disk, the present theorems show that any convex shape—whether a polygon, an infinite cone, or an open half‑plane—can serve as the “distance unit” without breaking planarity. This opens the door to a variety of applications: geometric spanners based on anisotropic distance functions, sensor‑network routing where communication ranges are convex but not circular, and motion‑planning regions defined by convex obstacles.

In the concluding section the authors outline future research directions, including dynamic algorithms that maintain G₁ or G₂ under point insertions and deletions, extensions to higher dimensions (where convexity alone no longer guarantees planarity), and the investigation of non‑convex shapes that still admit planar Delaunay‑like graphs under additional constraints.

Overall, the work demonstrates that convexity alone is the essential geometric condition for both translation‑only and scaled‑translation Delaunay‑type graphs to be planar, thereby significantly broadening the scope of planar geometric graph theory.