Impossible patterns of sphere tangencies

My latest preprint, “Tangent spheres and integer distances” (arXiv:2606.18569, to appear at CCCG), involves the patterns of external tangencies of circles, spheres or higher-dimensional hyperspheres. You can make a graph whose vertices are a given set of spheres and whose edges are pairs of externally-tangent spheres, and I’d like to understand which graphs are possible. By the circle packing…