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Minimal embeddings in the projective plane

✍ Scribed by Randby, Scott P.

Publisher: John Wiley and Sons
Year: 1997
Tongue: English
Weight: 161 KB
Volume: 25
Category: Article
ISSN: 0364-9024
DOI: 10.1002/(sici)1097-0118(199706)25:2<153::aid-jgt7>3.0.co;2-l

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✦ Synopsis

We show that if G is a graph embedded on the projective plane in such a way that each noncontractible cycle intersects G at least n times and the embedding is minimal with respect to this property (i.e., the representativity of the embedding is n), then G can be reduced by a series of reduction operations to an n × n × n projective grid. The reduction operations consist of changing a triangle of G to a triad, changing a triad of G to a triangle, and several others. We also show that if every proper minor of the embedding has representativity < n (i.e., the embedding is minimal), then G can be obtained from an n × n × n projective grid by a series of the two reduction operations described above. Hence every minimal embedding has the same number of edges.

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