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Embedding of planar graphs in the plane

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Book cover Ten Applications of Graph Theory

Part of the book series: Mathematics and Its Applications ((MAEE,volume 7))

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Abstract

In practice, there often arises the problem of drawing in the plane graphs which exist, say, in the form of electric networks or time schedules, without any crossing. Such an embedding without crossing is important for the design of printed circuits. If all of the components of an electric network are left unconsidered, the graph G of the network remains and the embedding of this yields a solution to the problem.

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Bibliography

  1. Auslander, L., and S. V. Parter: On imbedding graphs in the sphere, J. Math. Mech. 10 (1961), 517–523.

    MathSciNet  MATH  Google Scholar 

  2. Bader, W.: Das topologische Problem der gedruckten Schaltung und seine Lösung. Arch. Elektrotechnik 49 (1964), 2–12.

    Article  Google Scholar 

  3. Berge, C., und A. Ghouila-Houri: Programme, Spiele, Transportnetze, 2nd edn., Leipzig 1969. (Translated from French).

    Google Scholar 

  4. Dambitis, J. J.: Methode zur Konstruktion ebener Graphen (Russian), Latv. matem. ežegodnik 6 (1969), 41–63.

    MathSciNet  MATH  Google Scholar 

  5. Demoucron, G., Y. Malgrange et R. Pertuiset: Graphes planaires: Reconnaissance et construction de représentations planaires topologiques, Rev. Franç. Recherche Operat. 8, Nr. 30 (1964), 33–47

    Google Scholar 

  6. Fischer, G. J., and O. Wing: An algorithm for testing planar graphs from the incidence matrix, in: Proc. 7th Midwest Symp. on Circuit Theory, Ann Arbor, Mich., May 1964.

    Google Scholar 

  7. Fischer, G. J., and Wing, O: On correspondence between a class of planar graphs and bipartite graphs, IEEE Trans. Circuit Theory CT-12 (1965), 266–267.

    Google Scholar 

  8. Guillemin, E. A.: How to grow your trees from given cut-set or tie-set matrices, IRE Trans. Circuit Theory CT-6, Spec. Suppl. (1959), 110–126.

    Google Scholar 

  9. Halin, E.: Bemerkungen über ebene Graphen, Math. Ann. 153 (1964), 38–46.

    Article  MathSciNet  Google Scholar 

  10. Halin, R.: Über simpliziale Zerfällung n beliebiger (endlicher oder unendlicher) Graphen, Math. Ann. 156 (1964), 216–225.

    Article  MathSciNet  MATH  Google Scholar 

  11. Harary, F., and W. T. Tutte: A dual form of Kuratowski’s theorem, Canad. Math. Bull. 8 (1965), 17–20.

    Article  MathSciNet  Google Scholar 

  12. Hotz, G.: Einbettung von Streckenkomplexen in die Ebene, Math. Ann. 167 (1966), 214–223.

    Article  MathSciNet  MATH  Google Scholar 

  13. Jung, H.A.: Eine Verallgemeinerung des n-fachen Zusammenhanges für Graphen, Math. Ann. 187 (1970), 95–103.

    Article  MathSciNet  MATH  Google Scholar 

  14. Kuratowski, C.: Sur 1e problème des courves gauches en Topologie, Fund. Math. 15 (1930), 271–283.

    MATH  Google Scholar 

  15. Lempel, A., S. Even and I. Cederbaum: An algorithm for planarity testing of graphs, in: Théorie des Graphes, Paris/New York 1967, p. 215–232.

    Google Scholar 

  16. Lin, P. M.: On methods of detecting planar graphs, in: Proc. 8th Midwest Symp. on Circuit Theory, Colorado State Univ., June 1965, p. 14-15.

    Google Scholar 

  17. MacLane, S.: A structural characterisation of planar combinatorial graphs, Duke Math. J. 3 (1937), 340–472.

    Article  MathSciNet  Google Scholar 

  18. MacLane, S.: A combinatorial condition for planar graphs, Fund. Math. 28 (1937), 22–32.

    MATH  Google Scholar 

  19. Mayeda, W.: Necessary and sufficient conditions for realizability of cut-set matrices. IRE Trans. CT-7 (1969), 79–81.

    MathSciNet  Google Scholar 

  20. Sachs, H.: Einführung in die Theorie der endlichen Graphen, Teil I, Leipzig 1970.

    Google Scholar 

  21. Sachs, H.: Einführung in die Theorie der endlichen Graphen, Teil II, Leipzig 1972.

    Google Scholar 

  22. Tutte, W. T.: How to draw a graph, Proc. London Math. Soc. 13 (1963), 743–767.

    Article  MathSciNet  MATH  Google Scholar 

  23. Tutte, W. T.: A theorem on planar graphs, Trans. Amer. Math. Soc. 82 (1956), 99–116.

    Article  MathSciNet  MATH  Google Scholar 

  24. Tutte, W. T.: Separation of vertices by a circuit, Department of Combinatorics and Optimization, Research Report CORR 74-18, University of Waterloo, Ontario, 1974.

    Google Scholar 

  25. Wagner, K.: Über eine Eigenschaft der ebenen Komplexe, Math. Ann. 114 (1937), 570–590.

    Article  MathSciNet  Google Scholar 

  26. Wagner, K.: Eine Klasse minimaler nichtplättbarer Graphen, Math. Ann. 187 (1970), 104–113.

    Article  MathSciNet  MATH  Google Scholar 

  27. Whitney, H.: Non-separable and planar graphs, Trans. Amer. Math. Soc. 34 (1932), 339–362.

    Article  MathSciNet  Google Scholar 

  28. Whitney, H.: Planar graphs, Fund. Math. 21 (1933), 73–84.

    Google Scholar 

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© 1984 VEB Deutscher Verlag der Wissenschaften, Berlin

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Walther, H. (1984). Embedding of planar graphs in the plane. In: Ten Applications of Graph Theory. Mathematics and Its Applications, vol 7. Springer, Dordrecht. https://doi.org/10.1007/978-94-009-7154-7_11

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  • DOI: https://doi.org/10.1007/978-94-009-7154-7_11

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-94-009-7156-1

  • Online ISBN: 978-94-009-7154-7

  • eBook Packages: Springer Book Archive

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