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 addition of { \it some} $\ell$ edges. These graphs are identified with classes of { \it sparse} graphs; the results are also given in matroidal terms.
Categories: math.CO