Characterizing Sparse Graphs by Map Decompositions

Abstract

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.