Decomposition of graphs on surfaces (Q1369657)

For \(G= (V,E)\) an Eulerian graph imbedded on a triangulizable surface \(S\), \(\text{mincr}(G, D)\) denotes the minimum number of intersections of \(G\) and \(D'\) (counting multiplicities), where \(D'\) ranges over all closed curves freely homotopic to \(D\) and not intersecting \(V\). Also, \(\text{mincr}(C,D)\) denotes the minimum number of intersections of \(C'\) and \(D'\) (counting multiplicities), where \(C'\) and \(D'\) range over all closed curves freely homotopic to \(C\) and \(D\), respectively. The authors show that \(E\) can be decomposed into closed curves \(C_1,C_2,\dots,C_k\) such that \(\text{mincr}(G,D)= \sum^k_{i=1}\text{mincr}(C_i, D)\), for each closed curve \(D\) on \(S\). They also present two corollaries, one for bipartite graphs and one for homotopic circulations.