Graphs

Chapter
Part of the Algorithms and Combinatorics book series (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.

Keywords

Undirected Graph Topological Order Adjacency List Planar Embedding Acyclic Digraph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Berge, C. [1985]: Graphs. Second Edition. Elsevier, Amsterdam 1985MATHGoogle Scholar
  2. Bollobás, B. [1998]: Modern Graph Theory. Springer, New York 1998CrossRefMATHGoogle Scholar
  3. 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 1995Google Scholar
  4. Bondy, J.A., and Murty, U.S.R. [2008]: Graph Theory. Springer, New York 2008CrossRefMATHGoogle Scholar
  5. Diestel, R. [2010]: Graph Theory. Fourth Edition. Springer, New York 2010CrossRefGoogle Scholar
  6. Wilson, R.J. [2010]: Introduction to Graph Theory. Fifth Edition. Addison-Wesley, Reading 2010Google Scholar
  7. 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), 86–87MATHMathSciNetGoogle Scholar
  8. Camion, P. [1959]: Chemins et circuits hamiltoniens des graphes complets. Comptes Rendus Hebdomadaires des Séances de l’Académie des Sciences (Paris) 249 (1959), 2151–2152MATHMathSciNetGoogle Scholar
  9. Camion, P. [1968]: Modulaires unimodulaires. Journal of Combinatorial Theory A 4 (1968), 301–362CrossRefMATHMathSciNetGoogle Scholar
  10. Dirac, G.A. [1952]: Some theorems on abstract graphs. Proceedings of the London Mathematical Society 2 (1952), 69–81CrossRefMATHMathSciNetGoogle Scholar
  11. 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. 185–204Google Scholar
  12. Euler, L. [1736]: Solutio problematis ad geometriam situs pertinentis. Commentarii Academiae Petropolitanae 8 (1736), 128–140Google Scholar
  13. Euler, L. [1758]: Demonstratio nonnullarum insignium proprietatum quibus solida hedris planis inclusa sunt praedita. Novi Commentarii Academiae Petropolitanae 4 (1758), 140–160Google Scholar
  14. Hierholzer, C. [1873]: Über die Möglichkeit, einen Linienzug ohne Wiederholung und ohne Unterbrechung zu umfahren. Mathematische Annalen 6 (1873), 30–32CrossRefMathSciNetGoogle Scholar
  15. Hopcroft, J.E., and Tarjan, R.E. [1974]: Efficient planarity testing. Journal of the ACM 21 (1974), 549–568CrossRefMATHMathSciNetGoogle Scholar
  16. Kahn, A.B. [1962]: Topological sorting of large networks. Communications of the ACM 5 (1962), 558–562CrossRefMATHGoogle Scholar
  17. 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
  18. Knuth, D.E. [1968]: The Art of Computer Programming; Vol. 1. Fundamental Algorithms. Addison-Wesley, Reading 1968 (third edition: 1997)Google Scholar
  19. König, D. [1916]: Über Graphen und Ihre Anwendung auf Determinantentheorie und Mengenlehre. Mathematische Annalen 77 (1916), 453–465CrossRefMATHMathSciNetGoogle Scholar
  20. König, D. [1936]: Theorie der endlichen und unendlichen Graphen. Teubner, Leipzig 1936; reprint: Chelsea Publishing Co., New York 1950Google Scholar
  21. Kuratowski, K. [1930]: Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15 (1930), 271–283MATHGoogle Scholar
  22. Legendre, A.M. [1794]: Éléments de Géométrie. Firmin Didot, Paris 1794Google Scholar
  23. Minty, G.J. [1960]: Monotone networks. Proceedings of the Royal Society of London A 257 (1960), 194–212CrossRefMATHMathSciNetGoogle Scholar
  24. 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. 285–292Google Scholar
  25. Rédei, L. [1934]: Ein kombinatorischer Satz. Acta Litt. Szeged 7 (1934), 39–43MATHGoogle Scholar
  26. Robbins, H.E. [1939]: A theorem on graphs with an application to a problem of traffic control. American Mathematical Monthly 46 (1939), 281–283CrossRefMathSciNetGoogle Scholar
  27. Robertson, N., and Seymour, P.D. [1986]: Graph minors II: algorithmic aspects of tree-width. Journal of Algorithms 7 (1986), 309–322CrossRefMATHMathSciNetGoogle Scholar
  28. Robertson, N., and Seymour, P.D. [2004]: Graph minors XX: Wagner’s conjecture. Journal of Combinatorial Theory B 92 (2004), 325–357CrossRefMATHMathSciNetGoogle Scholar
  29. Tarjan, R.E. [1972]: Depth first search and linear graph algorithms. SIAM Journal on Computing 1 (1972), 146–160CrossRefMATHMathSciNetGoogle Scholar
  30. Thomassen, C. [1980]: Planarity and duality of finite and infinite graphs. Journal of Combinatorial Theory B 29 (1980), 244–271CrossRefMATHMathSciNetGoogle Scholar
  31. Thomassen, C. [1981]: Kuratowski’s theorem. Journal of Graph Theory 5 (1981), 225–241CrossRefMATHMathSciNetGoogle Scholar
  32. Tutte, W.T. [1961]: A theory of 3-connected graphs. Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen A 64 (1961), 441–455MATHMathSciNetGoogle Scholar
  33. Wagner, K. [1937]: Über eine Eigenschaft der ebenen Komplexe. Mathematische Annalen 114 (1937), 570–590CrossRefMathSciNetGoogle Scholar
  34. Whitney, H. [1932]: Non-separable and planar graphs. Transactions of the American Mathematical Society 34 (1932), 339–362CrossRefMathSciNetGoogle Scholar
  35. Whitney, H. [1933]: Planar graphs. Fundamenta Mathematicae 21 (1933), 73–84Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  1. 1.Research Institute for Discrete MathematicsUniversity of BonnBonnGermany

Personalised recommendations