  Sparse graphs and their associated matroids play an important role in
rigidity theory, where they capture the combinatorics of generically rigid
structures. We define a new family called {\bf graded sparse graphs}, arising
from generically pinned (completely immobilized) bar-and-joint frameworks and
prove that they also form matroids.
  We address five problems on graded sparse graphs: {\bf Decision}, {\bf
Extraction}, {\bf Components}, {\bf Optimization}, and {\bf Extension}. We
extend our {\bf pebble game algorithms} to solve them.
