Planar Graph Drawing Program Frye Theorem

In mathematics, Fáry's theorem states that whatsoever simple planar graph tin can be drawn without crossings so that its edges are direct line segments. That is, the ability to depict graph edges as curves instead of as straight line segments does not allow a larger grade of graphs to be drawn. The theorem is named after István Fáry, although it was proved independently past Klaus Wagner (1936), Fáry (1948), and Sherman Thou. Stein (1951).

Proof [edit]

Induction step for proof of Fáry's theorem.

I way of proving Fáry's theorem is to use mathematical induction.^[i] Let G be a simple plane graph with n vertices; we may add edges if necessary then that G is a maximally plane graph. If north < 3, the issue is trivial. If n ≥ 3, then all faces of K must be triangles, as we could add together an edge into any confront with more sides while preserving planarity, contradicting the supposition of maximal planarity. Choose some three vertices a, b, c forming a triangular face of G. We show past consecration 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 due north = three and a, b and c are the only vertices in G. Thus, we may presume that n ≥ four.

Past Euler'south formula for planar graphs, G has 3northward − 6 edges; equivalently, if one defines the deficiency of a vertex v in G to exist half dozen − deg(5), the sum of the deficiencies is 12. Since G has at least iv vertices and all faces of G are triangles, it follows that every vertex in Chiliad has degree at least three. Therefore each vertex in Thou has deficiency at most three, so there are at least four vertices with positive deficiency. In particular we can cull a vertex v with at virtually five neighbors that is different from a, b and c. Let G' be formed by removing v from G and retriangulating the face up f formed by removing 5. Past induction, G' has a combinatorially isomorphic straight line re-embedding in which abc is the outer face. Because the re-embedding of G' was combinatorially isomorphic to G', removing from it the edges which were added to create Thousand' leaves the face up f, which is at present a polygon P with at most five sides. To complete the drawing to a straight-line combinatorially isomorphic re-embedding of Thou, v should be placed in the polygon and joined by straight lines to the vertices of the polygon. By the art gallery theorem, there exists a point interior to P at which v can exist placed and so that the edges from v to the vertices of P do not cross any other edges, completing the proof.

The induction footstep of this proof is illustrated at right.

[edit]

De Fraysseix, Pach and Pollack showed how to find in linear time a direct-line cartoon in a filigree with dimensions linear in the size of the graph, giving a universal point set with quadratic size. A like method has been followed by Schnyder to prove enhanced bounds and a characterization of planarity based on the incidence partial gild. His work stressed the existence of a particular sectionalization of the edges of a maximal planar graph into 3 trees known every bit a Schnyder wood.

Tutte's jump theorem states that every 3-connected planar graph tin be drawn on a aeroplane without crossings and then that its edges are straight line segments and an outside face is a convex polygon (Tutte 1963). It is then called because such an embedding can be institute every bit the equilibrium position for a arrangement of springs representing the edges of the graph.

Steinitz's theorem states that every three-continued planar graph tin can be represented every bit the edges of a convex polyhedron in 3-dimensional space. A straight-line embedding of ${\displaystyle Thou,}$ of the blazon described past Tutte'due south theorem, may be formed by projecting such a polyhedral representation onto the aeroplane.

The Circle packing theorem states that every planar graph may be represented as the intersection graph of a collection of non-crossing circles in the aeroplane. Placing each vertex of the graph at the eye of the respective circumvolve leads to a straight line representation.

Unsolved problem in mathematics:

Does every planar graph accept a direct line representation in which all edge lengths are integers?

Heiko Harborth raised the question of whether every planar graph has a straight line representation in which all edge lengths are integers.^[two] The truth of Harborth's conjecture remains unknown as of 2014^[update]. Withal, integer-altitude straight line embeddings are known to be for cubic graphs.^[3]

Sachs (1983) raised the question of whether every graph with a linkless embedding in three-dimensional Euclidean space has a linkless embedding in which all edges are represented by straight line segments, analogously to Fáry'south theorem for ii-dimensional embeddings.

Run across likewise [edit]

• Bend minimization

Notes [edit]

1. ^ The proof that follows tin can exist found in Chartrand, Gary; Lesniak, Linda; Zhang, Ping (2010), Graphs & Digraphs (5th ed.), CRC Press, pp. 259–260, ISBN9781439826270 .
2. ^ Harborth et al. (1987); Kemnitz & Harborth (2001); Mohar & Thomassen (2001); Mohar (2003).
3. ^ Geelen, Guo & McKinnon (2008).

References [edit]

• Fáry, István (1948), "On directly-line representation of planar graphs", Acta Sci. Math. (Szeged), 11: 229–233, MR 0026311 .
• de Fraysseix, Hubert; Pach, János; Pollack, Richard (1988), "Small sets supporting Fary embeddings of planar graphs", Twentieth Almanac ACM Symposium on Theory of Calculating, pp. 426–433, doi:x.1145/62212.62254, ISBN0-89791-264-0, S2CID 15230919 .
• de Fraysseix, Hubert; Pach, János; Pollack, Richard (1990), "How to draw a planar graph on a grid", Combinatorica, 10: 41–51, doi:ten.1007/BF02122694, MR 1075065, S2CID 6861762 .
• Geelen, Jim; Guo, Anjie; McKinnon, David (2008), "Straight line embeddings of cubic planar graphs with integer border lengths" (PDF), J. Graph Theory, 58 (3): 270–274, doi:ten.1002/jgt.20304 .
• Harborth, H.; Kemnitz, A.; Moller, M.; Sussenbach, A. (1987), "Ganzzahlige planare Darstellungen der platonischen Korper", Elem. Math., 42: 118–122 .
• Kemnitz, A.; Harborth, H. (2001), "Plane integral drawings of planar graphs", Discrete Math., 236 (i–iii): 191–195, doi:10.1016/S0012-365X(00)00442-8 .
• Mohar, Bojan (2003), Problems from the volume Graphs on Surfaces .
• Mohar, Bojan; Thomassen, Carsten (2001), Graphs on Surfaces, Johns Hopkins Academy Press, pp. roblem 2.8.15, ISBN0-8018-6689-8 .
• Sachs, Horst (1983), "On a spatial analogue of Kuratowski's theorem on planar graphs — An open trouble", in Horowiecki, M.; Kennedy, J. W.; Sysło, M. M. (eds.), Graph Theory: Proceedings of a Conference held in Łagów, Poland, February x–13, 1981, Lecture Notes in Mathematics, vol. 1018, Springer-Verlag, pp. 230–241, doi:x.1007/BFb0071633, ISBN978-three-540-12687-4 .
• Schnyder, Walter (1990), "Embedding planar graphs on the grid", Proc. 1st ACM/SIAM Symposium on Discrete Algorithms (SODA), pp. 138–148, ISBN9780898712513 .
• Stein, Due south. K. (1951), "Convex maps", Proceedings of the American Mathematical Guild, 2 (three): 464–466, doi:x.2307/2031777, JSTOR 2031777, MR 0041425 .
• Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Society, 13: 743–767, doi:ten.1112/plms/s3-thirteen.one.743, MR 0158387 .
• Wagner, Klaus (1936), "Bemerkungen zum Vierfarbenproblem", Jahresbericht der Deutschen Mathematiker-Vereinigung, 46: 26–32 .

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Source: https://en.wikipedia.org/wiki/F%C3%A1ry%27s_theorem

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