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
General Literature
Berge , C. [1985]: Graphs. Second Edition. Elsevier, Amsterdam 1985
Bollobás , B. [1998]: Modern Graph Theory. Springer, New York 1998
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
Bondy , J.A., and Murty , U.S.R. [2008]: Graph Theory. Springer, New York 2008
Diestel , R. [2010]: Graph Theory. Fourth Edition. Springer, New York 2010
Wilson , R.J. [2010]: Introduction to Graph Theory. Fifth Edition. Addison-Wesley, Reading 2010
Cited References
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
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
Camion , P. [1968]: Modulaires unimodulaires. Journal of Combinatorial Theory A 4 (1968), 301–362
Dirac , G.A. [1952]: Some theorems on abstract graphs. Proceedings of the London Mathematical Society 2 (1952), 69–81
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
Euler , L. [1736]: Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Petropolitanae 8 (1736), 128–140
Euler , L. [1758]: Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Petropolitanae 4 (1758), 140–160
Hierholzer , C. [1873]: Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen 6 (1873), 30–32
Hopcroft , J.E., and Tarjan , R.E. [1974]: Efficient planarity testing. Journal of the ACM 21 (1974), 549–568
Kahn , A.B. [1962]: Topological sorting of large networks. Communications of the ACM 5 (1962), 558–562
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]
Knuth , D.E. [1968]: The Art of Computer Programming; Vol. 1. Fundamental Algorithms. Addison-Wesley, Reading 1968 (third edition: 1997)
König , D. [1916]: Über Graphen und ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465
König , D. [1936]: Theorie der endlichen und unendlichen Graphen. Teubner, Leipzig 1936; reprint: Chelsea Publishing Co., New York 1950
Kuratowski , K. [1930]: Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15 (1930), 271–283
Legendre , A.M. [1794]: Éléments de Géométrie. Firmin Didot, Paris 1794
Minty , G.J. [1960]: Monotone networks. Proceedings of the Royal Society of London A 257 (1960), 194–212
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
Rédei , L. [1934]: Ein kombinatorischer Satz. Acta Litt. Szeged 7 (1934), 39–43
Robbins , H.E. [1939]: A theorem on graphs with an application to a problem of traffic control. American Mathematical Monthly 46 (1939), 281–283
Robertson , N., and Seymour , P.D. [1986]: Graph minors II: algorithmic aspects of tree-width. Journal of Algorithms 7 (1986), 309–322
Robertson , N., and Seymour , P.D. [2004]: Graph minors XX: Wagner ’s conjecture. Journal of Combinatorial Theory B 92 (2004), 325–357
Tarjan , R.E. [1972]: Depth first search and linear graph algorithms. SIAM Journal on Computing 1 (1972), 146–160
Thomassen , C. [1980]: Planarity and duality of finite and infinite graphs. Journal of Combinatorial Theory B 29 (1980), 244–271
Thomassen , C. [1981]: Kuratowski ’s theorem. Journal of Graph Theory 5 (1981), 225–241
Tutte , W.T. [1961]: A theory of 3-connected graphs. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen A 64 (1961), 441–455
Wagner , K. [1937]: Über eine Eigenschaft der ebenen Komplexe. Mathematische Annalen 114 (1937), 570–590
Whitney , H. [1932]: Non-separable and planar graphs. Transactions of the American Mathematical Society 34 (1932), 339–362
Whitney , H. [1933]: Planar graphs. Fundamenta Mathematicae 21 (1933), 73–84
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Korte, B., Vygen, J. (2018). Graphs. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-56039-6_2
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DOI: https://doi.org/10.1007/978-3-662-56039-6_2
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