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.
