Abstract
We give two combinatorial characterizations of orientation reversing polygons in graphs embedded on surfaces (r-polygons) and use the notion of skew embedding introduced in [7] to characterize “parity” embeddings: an embedding has its odd polygons coinciding with itsr-polygons if and only if the skew embedding is in an orientable surface. The concept of “imbalance”, central for the proof, does not seem to appear explicitly before in the literature. Possible algorithmic implications of the parity embedding theorem are briefly discussed.
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Lins, S. Combinatorics of orientation reversing polygons. Aeq. Math. 29, 123–131 (1985). https://doi.org/10.1007/BF02189819
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DOI: https://doi.org/10.1007/BF02189819