  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.
