Planar Graph Drawing Straight Line Theorem

In mathematics, Fáry's theorem states that any simple planar graph can exist drawn without crossings so that its edges are directly line segments. That is, the power to draw graph edges equally curves instead of every bit straight line segments does not allow a larger class of graphs to exist fatigued. The theorem is named after István Fáry, although information technology was proved independently by Klaus Wagner (1936), Fáry (1948), and Sherman K. Stein (1951).

Proof [edit]

Induction pace for proof of Fáry's theorem.

One way of proving Fáry's theorem is to utilize mathematical induction.[1] Let G be a simple plane graph with n vertices; we may add edges if necessary and so that G is a maximally plane graph. If n < iii, the result is footling. If n ≥ three, then all faces of G must be triangles, as we could add together an edge into any face with more than sides while preserving planarity, contradicting the supposition of maximal planarity. Choose some three vertices a, b, c forming a triangular face up of G. We show by consecration on due north that in that location exists a straight-line combinatorially isomorphic re-embedding of One thousand in which triangle abc is the outer face up of the embedding. (Combinatorially isomorphic means that the vertices, edges, and faces in the new cartoon can be made to correspond to those in the old drawing, such that all incidences between edges, vertices, and faces—not but between vertices and edges—are preserved.) As a base instance, the result is trivial when northward = iii and a, b and c are the only vertices in G. Thus, we may assume that n ≥ 4.

By Euler's formula for planar graphs, G has 3n − 6 edges; equivalently, if one defines the deficiency of a vertex v in Thousand to be six − deg(v), the sum of the deficiencies is 12. Since G has at to the lowest degree four vertices and all faces of Thou are triangles, it follows that every vertex in G has degree at least three. Therefore each vertex in Yard has deficiency at about 3, so there are at least 4 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 five from G and retriangulating the face f formed by removing 5. By 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 Grand', removing from it the edges which were added to create G' leaves the face f, which is at present a polygon P with at most five sides. To consummate the drawing to a straight-line combinatorially isomorphic re-embedding of One thousand, v should be placed in the polygon and joined past straight lines to the vertices of the polygon. By the art gallery theorem, in that location exists a point interior to P at which five tin 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]

De Fraysseix, Pach and Pollack showed how to discover in linear time a straight-line drawing in a grid with dimensions linear in the size of the graph, giving a universal point set with quadratic size. A similar method has been followed past Schnyder to prove enhanced bounds and a characterization of planarity based on the incidence fractional society. His work stressed the existence of a particular sectionalization of the edges of a maximal planar graph into 3 trees known as a Schnyder wood.

Tutte's spring theorem states that every 3-continued planar graph can be drawn on a plane without crossings so that its edges are direct line segments and an exterior face is a convex polygon (Tutte 1963). Information technology is and then chosen because such an embedding tin can be found as the equilibrium position for a organisation of springs representing the edges of the graph.

Steinitz's theorem states that every iii-connected planar graph can be represented as the edges of a convex polyhedron in three-dimensional space. A straight-line embedding of M , {\displaystyle G,} of the blazon described past Tutte's theorem, may exist formed by projecting such a polyhedral representation onto the plane.

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 circle leads to a direct line representation.

Unsolved trouble in mathematics:

Does every planar graph have a straight 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.[2] The truth of Harborth's theorize remains unknown every bit of 2014[update]. Even so, integer-distance straight line embeddings are known to exist for cubic graphs.[iii]

Sachs (1983) raised the question of whether every graph with a linkless embedding in three-dimensional Euclidean infinite has a linkless embedding in which all edges are represented by direct line segments, analogously to Fáry'southward theorem for 2-dimensional embeddings.

See too [edit]

  • Bend minimization

Notes [edit]

  1. ^ The proof that follows can be constitute 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 straight-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 Annual ACM Symposium on Theory of Computing, pp. 426–433, doi:10.1145/62212.62254, 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:10.1007/BF02122694, MR 1075065, S2CID 6861762 .
  • Geelen, Jim; Guo, Anjie; McKinnon, David (2008), "Straight line embeddings of cubic planar graphs with integer edge lengths" (PDF), J. Graph Theory, 58 (3): 270–274, doi:10.1002/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 (1–iii): 191–195, doi:x.1016/S0012-365X(00)00442-eight .
  • Mohar, Bojan (2003), Bug 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. West.; Sysło, M. M. (eds.), Graph Theory: Proceedings of a Conference held in Łagów, Poland, February x–thirteen, 1981, Lecture Notes in Mathematics, vol. 1018, Springer-Verlag, pp. 230–241, doi:ten.1007/BFb0071633, ISBN978-3-540-12687-4 .
  • Schnyder, Walter (1990), "Embedding planar graphs on the grid", Proc. 1st ACM/SIAM Symposium on Discrete Algorithms (SODA), pp. 138–148, ISBN9780898712513 .
  • Stein, S. K. (1951), "Convex maps", Proceedings of the American Mathematical Social club, ii (3): 464–466, doi:ten.2307/2031777, JSTOR 2031777, MR 0041425 .
  • Tutte, W. T. (1963), "How to draw a graph", Proceedings of the London Mathematical Social club, 13: 743–767, doi:x.1112/plms/s3-13.1.743, 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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