Skip to main content
SpringerLink
Log in
Book cover

Combinatorial Optimization pp 117–137Cite as

Spanning Trees and Arborescences

  • Chapter

  • 1028 Accesses

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

Abstract

Consider a telephone company that wants to rent a subset from an existing set of cables, each of which connects two cities. The rented cables should suffice to connect all cities and they should be as cheap as possible. It is natural to model the network by a graph: the vertices are the cities and the edges correspond to the cables. By Theorem 2.4 the minimal connected spanning subgraphs of a given graph are its spanning trees.

This is a preview of subscription content, log in via an institution.

Chapter
USD   29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD   74.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Learn about institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

General Literature

  • Ahuja, R.K., Magnanti, T.L., and Orlin, J.B. [1993]: Network Flows. Prentice-Hall, Engle-wood Cliffs 1993, Chapter 13

    MATH  Google Scholar 

  • Balakrishnan, V.K. [1995]: Network Optimization. Chapman and Hall, London 1995, Chapter 1

    MATH  Google Scholar 

  • Cormen, T.H., Leiserson, C.E., and Rivest, R.L. [1990]: Introduction to Algorithms. MIT Press, Cambridge 1990, Chapter 24

    MATH  Google Scholar 

  • Gondran, M., and Minoux, M. [1984]: Graphs and Algorithms. Wiley, Chichester 1984, Chapter 4

    MATH  Google Scholar 

  • Magnanti, T.L., and Wolsey, L.A. [1995]: Optimal trees. In: Handbooks in Operations Research and Management Science; Volume 7: Network Models (M.O. Ball, T.L. Magnanti, C.L. Monma, G.L. Nemhauser, eds.), Elsevier, Amsterdam 1995, pp. 503–616

    Google Scholar 

  • Tarjan, R.E. [1983]: Data Structures and Network Algorithms. SIAM, Philadelphia 1983, Chapter 6

    Book  Google Scholar 

Cited References

  • Bock, F.C. [1971]: An algorithm to construct a minimum directed spanning tree in a directed network. In: Avi-Itzak, B. (Ed.): Developments in Operations Research. Gordon and Breach, New York 1971, 29–44

    Google Scholar 

  • Borůvka, O. [1926a]: O jistém problému minimálním. Práca Moravské Přírodovědecké Spolnečnosti 3 (1926), 37–58

    Google Scholar 

  • Borůvka, O. [1926b]: Příspevěk k řešení otázky ekonomické stavby. Elektrovodních sítí. Elektrotechnicky Obzor 15 (1926), 153–154

    Google Scholar 

  • Cayley, A. [1889]: A theorem on trees. Quarterly Journal on Mathematics 23 (1889), 376–378

    Google Scholar 

  • Chazelle, B. [2000]: A minimum spanning tree algorithm with inverse-Ackermann type complexity. Journal of the ACM 47 (2000), 1028–1047

    Article  MathSciNet  MATH  Google Scholar 

  • Cheriton, D., and Tarjan, R.E. [1976]: Finding minimum spanning trees. SIAM Journal on Computing 5 (1976), 724–742

    Article  MathSciNet  MATH  Google Scholar 

  • Chu, Y., and Liu, T. [1965]: On the shortest arborescence of a directed graph. Scientia Sinica 4 (1965), 1396–1400; Mathematical Review 33, # 1245

    Google Scholar 

  • Diestel, R. [1997]: Graph Theory. Springer, New York 1997

    MATH  Google Scholar 

  • Dijkstra, E.W. [1959]: A note on two problems in connexion with graphs. Numerische Mathematik 1 (1959), 269–271

    Article  MathSciNet  MATH  Google Scholar 

  • Dixon, B., Rauch, M., and Tarjan, R.E. [1992]: Verification and sensitivity analysis of minimum spanning trees in linear time. SIAM Journal on Computing 21 (1992), 1184–1192

    Article  MathSciNet  MATH  Google Scholar 

  • Edmonds, J. [1967]: Optimum branchings. Journal of Research of the National Bureau of Standards B 71 (1967), 233–240

    MathSciNet  Google Scholar 

  • Edmonds, J. [1970]: Submodular functions, matroids and certain polyhedra. In: Combinatorial Structures and Their Applications; Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications 1969 (R. Guy, H. Hanani, N. Sauer, J. Schonheim, eds.), Gordon and Breach, New York 1970, pp. 69–87

    Google Scholar 

  • Edmonds, J. [1973]: Edge-disjoint branchings. In: Combinatorial Algorithms (R. Rustin, ed.), Algorithmic Press, New York 1973, pp. 91–96

    Google Scholar 

  • Frank, A. [1978]: On disjoint trees and arborescences. In: Algebraic Methods in Graph Theory; Colloquia Mathematica; Soc. J. Bolyai 25 (L. Lovasz, V.T. Sos, eds.), North-Holland, Amsterdam 1978, pp. 159–169

    Google Scholar 

  • Frank, A. [1979]: Covering branchings. Acta Sci. Math. 41 (1979), 77–82

    MATH  Google Scholar 

  • Fredman, M.L., and Tarjan, R.E. [1987]: Fibonacci heaps and their uses in improved network optimization problems. Journal of the ACM 34 (1987), 596–615

    Article  MathSciNet  Google Scholar 

  • Fredman, M.L., and Willard, D.E. [1994]: Trans-dichotomous algorithms for minimum spanning trees and shortest paths. Journal of Computer and System Sciences 48 (1994), 533–551

    Article  MathSciNet  MATH  Google Scholar 

  • Fulkerson, D.R. [1974]: Packing rooted directed cuts in a weighted directed graph. Mathematical Programming 6 (1974), 1–13

    Article  MathSciNet  MATH  Google Scholar 

  • Gabow, H.N. [1995]: A matroid approach to finding edge connectivity and packing arborescences. Journal of Computer and System Sciences 50 (1995), 259–273

    Article  MathSciNet  MATH  Google Scholar 

  • Gabow, H.N., Galil, Z., and Spencer, T. [1989]: Efficient implementation of graph algorithms using contraction. Journal of the ACM 36 (1989), 540–572

    Article  MathSciNet  Google Scholar 

  • Gabow, H.N., Galil, Z., Spencer, T., and Tarjan, R.E. [1986]: Efficient algorithms for finding minimum spanning trees in undirected and directed graphs. Combinatorica 6 (1986), 109–122

    Article  MathSciNet  MATH  Google Scholar 

  • Gabow, H.N., and Manu, K.S. [1998]: Packing algorithms for arborescences (and spanning trees) in capacitated graphs. Mathematical Programming B 82 (1998), 83–109

    MathSciNet  Google Scholar 

  • Jarník, V. [1930]: O jistém problému minimálním. Práca Moravské Přírodovědecké Společnosti 6 (1930), 57–63

    Google Scholar 

  • Karger, D., Klein, P.N., and Tarjan, R.E. [1995]: A randomized linear-time algorithm to find minimum spanning trees. Journal of the ACM 42 (1995), 321–328

    Article  MathSciNet  MATH  Google Scholar 

  • Karp, R.M. [1972]: A simple derivation of Edmonds’ algorithm for optimum branchings. Networks 1 (1972), 265–272

    Article  MathSciNet  MATH  Google Scholar 

  • King, V. [1995]: A simpler minimum spanning tree verification algorithm. Algorithmica 18 (1997)

    Google Scholar 

  • Knuth, D.E. [1992]: Axioms and hulls; LNCS 606. Springer, Berlin 1992

    Book  Google Scholar 

  • Korte, B., and Nešetřil, J. [2001]: Vojtěch Jarník’s work in combinatorial optimization. Discrete Mathematics 235 (2001), 1–17

    Article  MathSciNet  MATH  Google Scholar 

  • Kruskal, J.B. [1956]: On the shortest spanning subtree of a graph and the travelling salesman problem. Proceedings of the AMS 7 (1956), 48–50

    Article  MathSciNet  MATH  Google Scholar 

  • Lovász, L. [1976]: On two minimax theorems in graph. Journal of Combinatorial Theory B 21 (1976), 96–103

    Article  Google Scholar 

  • Nash-Williams, C.S.J.A. [1961]: Edge-disjoint spanning trees of finite graphs. Journal of the London Mathematical Society 36 (1961), 445–450

    Article  MathSciNet  MATH  Google Scholar 

  • Nash-Williams, C.S.J.A. [1964]: Decompositions of finite graphs into forests. Journal of the London Mathematical Society 39 (1964), 12

    Article  MathSciNet  MATH  Google Scholar 

  • Nešetřil, J., Milková, E., and Nešetřilová, H. [2001]: Otakar Borůvka on minimum spanning tree problem. Translation of both the 1926 papers, comments, history. Discrete Mathematics 233 (2001), 3–36

    Article  MathSciNet  MATH  Google Scholar 

  • Prim, R.C. [1957]: Shortest connection networks and some generalizations. Bell System Technical Journal 36 (1957), 1389–1401

    Google Scholar 

  • Prüfer, H. [1918]: Neuer Beweis eines Satzes über Permutationen. Arch. Math. Phys. 27 (1918), 742–744

    Google Scholar 

  • Shamos, M.I., and Hoey, D. [1975]: Closest-point problems. Proceedings of the 16th Annual IEEE Symposium on Foundations of Computer Science (1975), 151-162

    Google Scholar 

  • Tarjan, R.E. [1975]: Efficiency of a good but not linear set union algorithm. Journal of the ACM 22 (1975), 215–225

    Article  MathSciNet  MATH  Google Scholar 

  • Tutte, W.T. [1961]: On the problem of decomposing a graph into n connected factor. Journal of the London Mathematical Society 36 (1961), 221–230

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

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

    Bernhard Korte & Jens Vygen

Authors
  1. Bernhard Korte
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jens Vygen
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2002 Springer-Verlag Berlin Heidelberg

About this chapter

Cite this chapter

Korte, B., Vygen, J. (2002). Spanning Trees and Arborescences. In: Combinatorial Optimization. Algorithms and Combinatorics, vol 21. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-662-21711-5_6

Download citation

  • DOI: https://doi.org/10.1007/978-3-662-21711-5_6

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-662-21713-9

  • Online ISBN: 978-3-662-21711-5

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation