The cycle structure for directed graphs on surfaces

In this paper, the cycle structures for directed graphs on surfaces are studied. If G is a strongly connected graph, C is a .-contractible directed cycle of G, then both of Int(C,.) and Ext(C,.) are strongly connected graph; the dimension of cycles space of G is identified. If G is a strongly connected graph, then the structure of MCB in G is unique. Let G be a strongly connected graph, if G has been embedded in orientable surface S g with f w (G) . 2 (f w (G) is the face-width of G), then any cycle base of G must contain at least 2g noncontractible directed cycles; if G has been embedded in non-orientable surface N g , then any cycle base of G must contain at least g noncontractible directed cycles.