Combinatorial Oriented Maps

Source abstract

An orientable map is often presented as a realization of a finite connected graph G in an orientable surface so that the complementary domains of G , the “faces” of the map are topological open discs. This is not the definition to be used in the paper. But let us contemplate it for a while. On each edge of G we can recognize two opposite directed edges, or “darts”. Let θ be the permutation of the dart-set S that interchanges each dart with its opposite. The darts radiating from a vertex v occur in a definite cyclic order, fixed by a chosen positive sense of rotation on the surface. The cyclic orders at the various vertices are the cycles of a permutation P of S . The choice of P rather than P –l , which corresponds to the other sense of rotation, makes the map “oriented”.