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In combinatorial mathematics, rotation systems encode embeddings of graphs onto orientable surfaces, by describing the circular ordering of a graph's edges around each vertex. A more formal definition of a rotation system involves pairs of permutations; such a pair is sufficient to determine a multigraph, a surface, and a 2-cell embedding of the multigraph onto the surface. Every rotation scheme defines a unique 2-cell embedding of a connected multigraph on a closed oriented surface (up to orientation preserving topological equivalence). Conversely, any embedding of a connected multigraph ''G'' on an oriented closed surface defines a unique rotation system having ''G'' as its underlying multigraph. This fundamental equivalence between rotation systems and 2-cell-embeddings was first settled in a dual form by Heffter and extensively used by Ringel during the 1950s. Independently, Edmonds gave the primal form of the theorem and the details of his study have been popularized by Youngs. The generalization to the whole set of multigraphs was developed by Gross and Alpert. Rotation systems are related to, but not the same as, the ''rotation maps'' used by Reingold et al. (2002) to define the zig-zag product of graphs. A rotation system specifies a circular ordering of the edges around each vertex, while a rotation map specifies a (non-circular) permutation of the edges at each vertex. In addition, rotation systems can be defined for any graph, while as Reingold et al. define them rotation maps are restricted to regular graphs. == Formal definition == Formally, a rotation system is defined as a pair (σ,θ) where σ and θ are permutations acting on the same ground set ''B'', θ is a fixed-point-free involution, and the group <σ,θ> generated by σ and θ acts transitively on ''B''. To derive a rotation system from a 2-cell embedding of a connected multigraph ''G'' on an oriented surface, let ''B'' consist of the ''darts'' (or ''flags'', or ''half-edges'') of ''G''; that is, for each edge of ''G'' we form two elements of ''B'', one for each endpoint of the edge. Even when an edge has the same vertex as both of its endpoints, we create two darts for that edge. We let θ(''b'') be the other dart formed from the same edge as ''b''; this is clearly an involution with no fixed points. We let σ(''b'') be the dart in the clockwise position from ''b'' in the cyclic order of edges incident to the same vertex, where "clockwise" is defined by the orientation of the surface. If a multigraph is embedded on an orientable but not oriented surface, it generally corresponds to two rotation systems, one for each of the two orientations of the surface. These two rotation systems have the same involution θ, but the permutation σ for one rotation system is the inverse of the corresponding permutation for the other rotation system. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Rotation system」の詳細全文を読む スポンサード リンク
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