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Fáry's Theorem
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Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, Fáry''s theorem states that any simple planar graph can be drawn without crossings so that its edges are straight line segments. That is, the ability to draw graph edges as curves instead of as straight line segments does not allow a larger class of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently by Klaus Wagner (1936), Fáry (1948), and S. K. Stein (1951).Let G be a simple planar graph with n vertices; we may add edges if necessary so that G is maximal planar. All faces of G will be triangles, as we could add an edge into any face with more sides while preserving planarity, contradicting the assumption of maximal planarity. Choose some three vertices a,b,c forming a triangular face of G. We prove by induction on n that there exists a straight-line embedding of G in which triangle abc is the outer face of the embedding. As a base case, the result is trivial when n = 3 and a,b, and c are the only vertices in G. Otherwise, all vertices in G have at least three neighbors.
Weitere Informationen
- Allgemeine Informationen
- GTIN 09786131266027
- Editor Lambert M. Surhone, Miriam T. Timpledon, Susan F. Marseken
- Größe H220mm x B220mm
- EAN 9786131266027
- Format Fachbuch
- Titel Fáry's Theorem
- Herausgeber Betascript Publishing
- Anzahl Seiten 64
- Genre Mathematik
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