  We revisit the concept of minimal rigidity as applied to soft repulsive,
frictionless sphere packings in two-dimensions with the introduction of the
jamming graph. Minimal rigidity is a purely combinatorial property encoded via
Laman's theorem in two-dimensions. It constrains the global, average
coordination number of the graph, for example. However, minimal rigidity does
not address the geometry of local mechanical stability. The jamming graph
contains both properties of global mechanical stability at the onset of jamming
and local mechanical stability. We demonstrate how jamming graphs can be
constructed using local moves via the Henneberg construction such that these
graphs fall under the jurisdiction of correlated percolation. We then probe how
jamming graphs destabilize, or become unjammed, by deleting a bond and
computing the resulting rigid cluster distribution. We also study how the
system restabilizes with the addition of new contacts and how a jamming graph
with extra/redundant contacts destabilizes. The latter endeavor allows us to
probe a disc packing in the rigid phase and uncover a potentially new diverging
lengthscale associated with the random deletion of contacts as compared to the
study of cut-out (or frozen in) subsystems.
