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

Combinatorics;Metric Geometry