Fáry's theorem

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).

One way of proving Fáry's theorem is to use mathematical induction. Let G be a simple plane graph with n vertices; we may add edges if necessary so that G is a maximally plane graph. All faces of G must 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 combinatorially isomorphic re-embedding of G in which triangle abc is the outer face of the embedding. (Combinatorially isomorphic means that the vertices, edges, and faces in the new drawing can be made to correspond to those in the old drawing, such that all incidences between edges, vertices, and faces—not just between vertices and edges—are preserved.) 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.

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