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Ozkan et al. conjectured that any packing of $n$ spheres with generic radiiwill be stress-free, and hence will have at most $3n-6$ contacts. In this paperwe prove that this conjecture is true for any sphere packing with contact graphof the form $G \oplus K_2$, i.e., the graph formed by connecting every vertexin a graph $G$ to every vertex in the complete graph with two vertices. We alsoprove the converse of the conjecture holds in this special case: specifically,a graph $G \oplus K_2$ is the contact graph of a generic radii sphere packingif and only if $G$ is a penny graph with no cycles.

Identifying contact graphs of sphere packings with generic radii