Journal of Combinatorial Mathematics and Combinatorial Computing
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 ℓ edges; (2) the addition of some ℓ edges. These graphs are identified with classes of sparse graphs; the results are also given in matroidal terms.
Haas, Ruth; Lee, Audrey; Streinu, Ileana; and Theran, Louis, "Characterizing Sparse Graphs by Map Decompositions" (2007). Faculty Publications. Paper 4.