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.
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