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 sparse graphs; the results are also given in matroidal terms.