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Combinatorial Optimization pp 13–49Cite as

Graphs

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Part of the book series: Algorithms and Combinatorics ((AC,volume 21))

Abstract

Graphs are a fundamental combinatorial structure used throughout this book. In this chapter we not only review the standard definitions and notation, but also prove some basic theorems and mention some fundamental algorithms.

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References

  • Berge, C. [1985]: Graphs. Second Edition. Elsevier, Amsterdam 1985

    MATH  Google Scholar 

  • Bollobás, B. [1998]: Modern Graph Theory. Springer, New York 1998

    Book  MATH  Google Scholar 

  • Bondy, J.A. [1995]: Basic graph theory: paths and circuits. In: Handbook of Combinatorics; Vol. 1 (R.L. Graham, M. Grötschel, L. Lovász, eds.), Elsevier, Amsterdam 1995

    Google Scholar 

  • Bondy, J.A., and Murty, U.S.R. [2008]: Graph Theory. Springer, New York 2008

    Book  MATH  Google Scholar 

  • Diestel, R. [2010]: Graph Theory. Fourth Edition. Springer, New York 2010

    Book  Google Scholar 

  • Wilson, R.J. [2010]: Introduction to Graph Theory. Fifth Edition. Addison-Wesley, Reading 2010

    Google Scholar 

  • Aoshima, K., and Iri, M. [1977]: Comments on F. Hadlock’s paper: finding a maximum cut of a planar graph in polynomial time. SIAM Journal on Computing 6 (1977), 86–87

    MATH  MathSciNet  Google Scholar 

  • Camion, P. [1959]: Chemins et circuits hamiltoniens des graphes complets. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris) 249 (1959), 2151–2152

    MATH  MathSciNet  Google Scholar 

  • Camion, P. [1968]: Modulaires unimodulaires. Journal of Combinatorial Theory A 4 (1968), 301–362

    Article  MATH  MathSciNet  Google Scholar 

  • Dirac, G.A. [1952]: Some theorems on abstract graphs. Proceedings of the London Mathematical Society 2 (1952), 69–81

    Article  MATH  MathSciNet  Google Scholar 

  • Edmonds, J., and Giles, R. [1977]: A min-max relation for submodular functions on graphs. In: Studies in Integer Programming; Annals of Discrete Mathematics 1 (P.L. Hammer, E.L. Johnson, B.H. Korte, G.L. Nemhauser, eds.), North-Holland, Amsterdam 1977, pp. 185–204

    Google Scholar 

  • Euler, L. [1736]: Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Petropolitanae 8 (1736), 128–140

    Google Scholar 

  • Euler, L. [1758]: Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Petropolitanae 4 (1758), 140–160

    Google Scholar 

  • Hierholzer, C. [1873]: Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen 6 (1873), 30–32

    Article  MathSciNet  Google Scholar 

  • Hopcroft, J.E., and Tarjan, R.E. [1974]: Efficient planarity testing. Journal of the ACM 21 (1974), 549–568

    Article  MATH  MathSciNet  Google Scholar 

  • Kahn, A.B. [1962]: Topological sorting of large networks. Communications of the ACM 5 (1962), 558–562

    Article  MATH  Google Scholar 

  • Karzanov, A.V. [1970]: An efficient algorithm for finding all the bi-components of a graph. In: Trudy 3-ĭ Zimneĭ Shkoly po Matematicheskomu Programmirovaniyu i Smezhnym Voprosam (Drogobych, 1970), Issue 2, Moscow Institute for Construction Engineering (MISI) Press, Moscow, 1970, pp. 343–347 [in Russian]

    Google Scholar 

  • Knuth, D.E. [1968]: The Art of Computer Programming; Vol. 1. Fundamental Algorithms. Addison-Wesley, Reading 1968 (third edition: 1997)

    Google Scholar 

  • König, D. [1916]: Über Graphen und Ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465

    Article  MATH  MathSciNet  Google Scholar 

  • König, D. [1936]: Theorie der endlichen und unendlichen Graphen. Teubner, Leipzig 1936; reprint: Chelsea Publishing Co., New York 1950

    Google Scholar 

  • Kuratowski, K. [1930]: Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15 (1930), 271–283

    MATH  Google Scholar 

  • Legendre, A.M. [1794]: Éléments de Géométrie. Firmin Didot, Paris 1794

    Google Scholar 

  • Minty, G.J. [1960]: Monotone networks. Proceedings of the Royal Society of London A 257 (1960), 194–212

    Article  MATH  MathSciNet  Google Scholar 

  • Moore, E.F. [1959]: The shortest path through a maze. Proceedings of the International Symposium on the Theory of Switching; Part II. Harvard University Press 1959, pp. 285–292

    Google Scholar 

  • Rédei, L. [1934]: Ein kombinatorischer Satz. Acta Litt. Szeged 7 (1934), 39–43

    MATH  Google Scholar 

  • Robbins, H.E. [1939]: A theorem on graphs with an application to a problem of traffic control. American Mathematical Monthly 46 (1939), 281–283

    Article  MathSciNet  Google Scholar 

  • Robertson, N., and Seymour, P.D. [1986]: Graph minors II: algorithmic aspects of tree-width. Journal of Algorithms 7 (1986), 309–322

    Article  MATH  MathSciNet  Google Scholar 

  • Robertson, N., and Seymour, P.D. [2004]: Graph minors XX: Wagner’s conjecture. Journal of Combinatorial Theory B 92 (2004), 325–357

    Article  MATH  MathSciNet  Google Scholar 

  • Tarjan, R.E. [1972]: Depth first search and linear graph algorithms. SIAM Journal on Computing 1 (1972), 146–160

    Article  MATH  MathSciNet  Google Scholar 

  • Thomassen, C. [1980]: Planarity and duality of finite and infinite graphs. Journal of Combinatorial Theory B 29 (1980), 244–271

    Article  MATH  MathSciNet  Google Scholar 

  • Thomassen, C. [1981]: Kuratowski’s theorem. Journal of Graph Theory 5 (1981), 225–241

    Article  MATH  MathSciNet  Google Scholar 

  • Tutte, W.T. [1961]: A theory of 3-connected graphs. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen A 64 (1961), 441–455

    MATH  MathSciNet  Google Scholar 

  • Wagner, K. [1937]: Über eine Eigenschaft der ebenen Komplexe. Mathematische Annalen 114 (1937), 570–590

    Article  MathSciNet  Google Scholar 

  • Whitney, H. [1932]: Non-separable and planar graphs. Transactions of the American Mathematical Society 34 (1932), 339–362

    Article  MathSciNet  Google Scholar 

  • Whitney, H. [1933]: Planar graphs. Fundamenta Mathematicae 21 (1933), 73–84

    Google Scholar 

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Authors and Affiliations

  1. Research Institute for Discrete Mathematics, University of Bonn, Lennéstr. 2, 53113, Bonn, Germany

    Bernhard Korte & Jens Vygen

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  1. Bernhard Korte
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  2. Jens Vygen
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Correspondence to Bernhard Korte .

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© 2012 Springer-Verlag Berlin Heidelberg

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Korte, B., Vygen, J. (2012). Graphs. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-24488-9_2

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