Fáry's theorem

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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 same theorem was proven independently by Wagner, Fáry, and Stein. Fraysseix, Pach and Pollack showed how to find in linear time a straight-line drawing in a grid with dimensions linear in the size of the graph. A similar method has been followed by Schnyder to prove enhanced bounds and a characterization of planarity based on the incidence partial order. His work stressed the existence of a particular partition of the edges of a maximal planar graph into three trees known as a Schnyder wood.

[edit] Proof

Induction step for proof of Fáry's theorem
Induction step for proof of Fáry's theorem

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.

By Euler's Formula for planar graphs, G has 3n-6 edges; equivalently, if one defines the deficiency of a vertex v in G to be six minus the degree of v, the sum of the deficiencies is twelve. Each vertex in G can have deficiency at most three, so there are at least four vertices with positive deficiency. In particular we can choose a vertex v with at most five neighbors that is different from a, b, and c. Let G' be formed by removing v from G and retriangulating the face formed by removing v. By induction, G' has a straight line embedding in which abc is the outer face. Remove the added edges in G', forming a polygon P with at most five sides into which v should be placed to complete the drawing. By the Art gallery theorem, there exists a point interior to P at which v can be placed so that the edges from v to the vertices of P do not cross any other edges, completing the proof.

The induction step of this proof is illustrated at right.

[edit] Related Results

Tutte's spring theorem states that every 3-connected planar graph can be drawn on a plane without crossings so that its edges are straight line segments and an outside face is a convex polygon (Tutte 1963). It is so called because such an embedding can be found as the equilibrium position for a system of springs representing the edges of the graph.

Steinitz's theorem states that every 3-connected planar graph can be represented as the edges of a convex polyhedron in three-dimensional space. A straight-line embedding of G, of the type described by Tutte's theorem, may be formed by projecting such a polyhedral representation onto the plane.

It follows from results of Koebe, Andreev, and Thurston (see, e.g., Stephenson 2003) that every planar graph may be represented as the intersection graph of a collection of non-crossing circles in the plane. Placing each vertex of the graph at the center of the corresponding circle leads to a straight line representation.

H. Harborth (Mohar and Thomassen 2001; Mohar 2003) raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers. The answer remains unknown as of 2006.

[edit] References

  • Fáry, István (1948). "On straight-line representation of planar graphs". Acta Sci. Math. (Szeged) 11: 229–233. MR0026311. 
  • de Fraysseix, Hubert, Pach, János, and Pollack, Richard (1988). "Small sets supporting Fary embeddings of planar graphs". Twentieth Annual ACM Symposium on Theory of Computing: 426–433. DOI:10.1145/62212.62254. 
  • de Fraysseix, Hubert, Pach, János, and Pollack, Richard (1990). "How to draw a planar graph on a grid". Combinatorica 10: 41–51. DOI:10.1007/BF02122694. MR1075065. 
  • Mohar, Bojan; Thomassen, Carsten (2001). Graphs on Surfaces. Johns Hopkins University Press, problem 2.8.15. ISBN 0801866898. 
  • Tutte, W. T. (1963). "How to draw a graph". Proceedings of the London Mathematical Society 13: 743–767. MR0158387. 
  • Wagner, Klaus (1936). "Bemerkungen zum Vierfarbenproblem". Jahresbericht. German. Math.-Verein. 46: 26–32. 
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