Minimal Graphs with Respect to Geometric Distance Realizability

Abstract

A graph G is minimal non-unit-distance graph if there is no drawing of G in Euclidean plane having all edges of unit length, but, for each edge e of G, G − e has such a drawing. We prove that, for infinitely many n, the number of non-isomorphic n-vertex minimal non-unit-distance graphs is at least exponential in n.

Keywords

• unit-distance graph
• odd-distance graph
• euclidean plane
• 05c62