Kronecker covers, V-construction, unit-distance graphs and isometric point-circle configurations

Abstract

We call a convex polytope P of dimension 3 admissible if it has the following two properties: (1) for each vertex of P the set of its first-neighbours is coplanar; (2) all planes determined by the first-neighbours are distinct. It is shown that the Levi graph of a point-plane configuration obtained by V-construction from an admissible polytope P is the Kronecker cover of the 1-skeleton of P. We investigate the combinatorial nature of the V-construction and use it on unit-distance graphs to construct novel isometric point-circle configurations. In particular, we present an infinite series all of whose members are subconfigurations of the renowned