Characterizing Sparse Graphs by Map Decompositions

A 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 any \(\ell\) edges; (2) the addition of some \(\ell\) edges. These graphs are identified with classes of \emph{sparse} graphs; the results are also given in matroidal terms.