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Embeddable of one-vertex graph with rotations into torus

by   Tim Berezin, et al.

We are interested in finding combinatorial criteria for embedding graphs into torus and using them in the embedding check algorithm. It is a well-known fact that any connected graph can be reduced to a one-vertex graph with loops and is homotopy equivalent to such a graph. If a connected graph has intersection of some edges, then some loops of one-vertex graph will also intersect. Therefore the problem of embedding graphs into some surface is equivalent to the question of embedding one-vertex graph in given surface. This paper considers a graph with rotations or rotation system called hieroglyph. The equivalence of four conditions is proved: (1) embedding hieroglyph into torus; (2) the absence of forbidden subgraphs in a hieroglyph; (3) some condition for the graph of loops; (4) the existence of a reduction of hieroglyph to one of the list of hieroglyphs. We also prove the existence of an algorithm with complexity O(n) that recognizes embedding hieroglyph into torus.


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