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 (com
pletely 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.
A {bf map} is a graph that admits an orientation of its edges so that each vertex has out-degree exactly 1. We characterize graphs which admit a decomposition into $k$ edge-disjoint maps after: (1) the addition of {it any} $ell$ edges; (2) the additi
on of {it some} $ell$ edges. These graphs are identified with classes of {it sparse} graphs; the results are also given in matroidal terms.
