Skip to main content

Introduction

  • Chapter
  • First Online:
Generalized Connectivity of Graphs

Part of the book series: SpringerBriefs in Mathematics ((BRIEFSMATH))

Abstract

In this section, we will bring together most of the terminology and notation used in this monograph. For those not given here, they will be defined whenever needed. All graphs considered in this book are finite, simple, and undirected, unless otherwise stated. We follow the graph theoretical terminology and notation of [19, 20] for all those not defined here.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Bibliography

  1. Barden, B., Libeskind-Hadas, R., Davis, J., Williams, W.: On edge-disjoint spanning trees in hypercubes. Inform. Proc. Lett. 70 (1), 13–16 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  2. Bauer, D., Broersma, H., Schmeichel, E.: Toughness in graphs: A survey. Graphs Combin. 22, 1–35 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beineke, L.W., Wilson, R.J.: Topics in Structural Graph Theory. Cambrige University Press, Cambrige (2013)

    MATH  Google Scholar 

  4. Beineke, L.W., Oellermann, O.R., Pippert, R.E.: The average connectivity of a graph. Discrete Math. 252 (1-3), 31–45 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  5. Boesch, F.T., Chen, S.: A generalization of line connectivity and optimally invulnerable graphs. SIAM J. Appl. Math. 34 (4), 657–665 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bollobás, B.: Extremal Graph Theory. Academic press, New York (1978)

    MATH  Google Scholar 

  7. Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, Amsterdam (1976)

    Book  MATH  Google Scholar 

  8. Bondy, J.A., Murty, U.S.R.: Graph Theory, GTM 244. Springer, New York (2008)

    Book  MATH  Google Scholar 

  9. Cai, Q., Li, X., Song, J.: Solutions to conjectures on the (k, ℓ)-rainbow index of complete graphs. Networks 62, 220–224 (2013)

    Google Scholar 

  10. Cai, Q., Li, X., Zhao, Y.: Note on the upper bound of the rainbow index of a graph. Discrete Appl. Math., http://dx.doi.org/10.1016/j.dam.2015.10.019 (in press)

  11. Catlin, P.A., Grossman, J.W., Hobbs, A.M., Lai, H.J.: Fractional arboricity, strength, and principal partitions in graphs and matroids. Discrete Appl. Math. 40 (3), 285–302 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  12. Chartrand, G., Kapoor, S.F., Lesniak, L., Lick, D.R.: Generalized connectivity in graphs. Bull. Bombay Math. Colloq. 2, 1–6 (1984)

    Google Scholar 

  13. Chartrand, G., Okamoto, F., Zhang, P.: Rainbow trees in graphs and generalized connectivity. Networks 55 (4), 360–367 (2010)

    MathSciNet  MATH  Google Scholar 

  14. Chen, L., Li, X., Liu, M., Mao, Y.: A solution to a conjecture on the generalized connectivity of graphs. J. Combin. Optim. (2015). DOI:10.1007/s10878-015-9955-x

    Google Scholar 

  15. Chen, X., Li, X., Lian, H.: Note on packing of edge-disjoint spanning trees in sparse random graphs. arXiv:1301.1097 [math.CO] (2013)

    Google Scholar 

  16. Chen, L., Li, X., Yang, K., Zhao, Y.: The 3-rainbow index of a graph. Discuss. Math. Graph Theory 35 (1), 81–94 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Cheng, X., Du, D.: Steiner Trees in Industry. Kluwer Academic Publisher, Dordrecht (2001)

    MATH  Google Scholar 

  18. Chvátal, V.: Tough graphs and hamiltonian circuits. Discrete Math. 5 (3), 215–228 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  19. Cunningham, W.H.: Optimal attack and reinforcement of a netwok. J. ACM 32 (3), 549–561 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  20. Day, D.P., Oellermann, O.R., Swart, H.C.: The ℓ-connectivity function of trees and complete multipartite graphs. J. Combin. Math. Combin. Comput. 10, 183–192 (1991)

    MathSciNet  MATH  Google Scholar 

  21. Dirac, G.A.: In abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. Math. Nach 22, 61–85 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  22. Du, D., Hu, X.: Steiner Tree Problems in Computer Communication Networks. World Scientific, Singapore (2008)

    Book  MATH  Google Scholar 

  23. Elias, P., Feinstein, A., Shannon, C.E.: A note on the maximum flow through a network. IRE Trans. Inform. Theory, IT-2 2 (4), 117–119 (1956)

    Google Scholar 

  24. Esfahanian, A.H.: Generalized measures of fault tolerance with application to N-cube networks. IEEE Trans. Comput. 38, 1586–1591 (1989)

    Article  Google Scholar 

  25. Ford, L.R., Fulkerson, D.R.: Maximum flow through a network. Can. J. Math. 8, 399–404 (1956)

    Article  MathSciNet  MATH  Google Scholar 

  26. Fragopoulou, P., Akl, S.G.: Edge-disjoint spanning trees on the star network with applications to fault tolerance. IEEE Trans. Comput. 45 (2), 174–185 (1996)

    Article  MATH  Google Scholar 

  27. Gao, P., Pérez-Giménez, X., Sato, C.M.: Arboricity and spanning-tree packing of random graphs. In: Proceeding SODA ’14 Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 317–326 (2014)

    Google Scholar 

  28. Goldsmith, D.L.: On the second-order edge-connectivity of a graph. Congressus Numerantium 29, 479–484 (1980)

    MathSciNet  Google Scholar 

  29. Goldsmith, D.L., Manvel, B., Faber, V.: Separation of graphs into three components by the removal of edges. J. Graph Theory 4 (2), 213–218 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  30. Grötschel, M.: The Steiner tree packing problem in VLSI design. Math. Program. 78, 265–281 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  31. Grötschel, M., Martin, A., Weismantel, R.: Packing Steiner trees: A cutting plane algorithm and commputational results. Math. Program. 72, 125–145 (1996)

    Article  MATH  Google Scholar 

  32. Gu, R., Li, X., Shi, Y.: The generalized 3-connectivity of random graphs. Acta Math. Sin. (Chinese Edition) 57 (2), 321–330 (2014)

    Google Scholar 

  33. Gusfield, D.: Connectivity and edge-disjoint spanning trees. Infor. Process. Lett. 16 (2), 87–89 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Hager, M.: Pendant tree-connectivity. J. Comb. Theory 38 (2), 179–189 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  35. Hager, M.: Path-connectivity in graphs. Discrete Math. 59 (1-2), 53–59 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)

    MATH  Google Scholar 

  37. Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. Inform. Comput. 79 (1), 43–59 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  38. Jain, K., Mahdian, M., Salavatipour, M.: Packing Steiner trees. In: Proc. 14th ACM-SIAM symposium on Discterte Algorithms, Baltimore, pp. 266–274 (2003)

    Google Scholar 

  39. Kriesell, M.: Edge-disjoint trees containing some given vertices in a graph. J. Combin. Theory Ser. B 88 (1), 53–65 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  40. Kriesell, M.: Local spanning trees in graphs and hypergraph decomposition with respect to edge-connectivity. Electron. Notes Discrete Math. 3, 110–113 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  41. Lau, L.: An approximate max-Steiner-tree-packing min-Steiner-cut theorem. Combinatorica 27, 71–90 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  42. Li, H., Li, X., Mao, Y.: On extremal graphs with at most two internally disjoint Steiner trees connecting any three vertices. Bull. Malays. Math. Sci. Soc. 37 (3), 747–756 (2014)

    MathSciNet  MATH  Google Scholar 

  43. Li, X., Mao, Y.: Nordhaus-Gaddum-type results for the generalized edge-connectivity of graphs. Discrete Appl. Math. 185, 102–112 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Li, X., Mao, Y.: The generalized 3-connectivity of lexicographic product graphs. Discrete Math. Theor. Comput. Sci. 16 (1), 339–354 (2014)

    MathSciNet  MATH  Google Scholar 

  45. Li, X., Mao, Y.: The minimal size of a graph with given generalized 3-edge-connectivity. Ars Combin. 118, 63–72 (2015)

    MathSciNet  MATH  Google Scholar 

  46. Li, X., Mao, Y., Sun, Y.: On the generalized (edge-)connectivity of graphs. Australasian J. Combin. 58 (2), 304–319 (2014)

    MathSciNet  MATH  Google Scholar 

  47. Li, X., Mao, Y.: Graphs with large generalized (edge-)connectivity. Discuss. Math. Graph Theory. Accepted by arXiv: 1305.1089 [math.CO] (2013)

    Google Scholar 

  48. Li, X., Schiermeyer, I., Yang, K., Zhao, Y.: Graphs with 3-rainbow index n − 1 and n − 2. Discuss. Math. Graph Theory 35 (1), 105–120 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  49. Li, X., Schiermeyer, I., Yang, K., Zhao, Y.: Graphs with 4-rainbow index 3 and n − 1. Discuss. Math. Graph Theory 35 (2), 387–398 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  50. Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: A survey. Graphs Combin. 29 (1), 1–38 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  51. Li, X., Sun, Y.: Rainbow Connections of Graphs. Springer Briefs in Math. Springer, New York (2012)

    Book  MATH  Google Scholar 

  52. Li, X., Yue, J., Zhao, Y.: The generalized 3-edge-connectivity of lexicographic product graphs, in Comb. Optim. Appl.. LNCS 8881 (Proc. COCOA2014, Maui, HI, USA), pp. 412–425 (2014)

    Google Scholar 

  53. Li, X., Zhao, Y.: On graphs with only one Steiner tree connecting any k vertices. arXiv:1301.4623v1 [math.CO] (2013)

    Google Scholar 

  54. Libeskind-Hadas, R., Mazzoni, D., Rajagopalan, R.: Tree-based multicasting in wormhole-routed irregular topologies. In: Proc. Merged 12th Int’l Parallel Processing Symp. and the Ninth Symp. Parallel and Distributed Processing, pp. 244–249, Apr. (1998)

    Google Scholar 

  55. Lü, M., Chen, G.L., Xu, J.: On super edge-connectivity of Cartesian product graphs. Networks 49 (2), 135–157 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  56. Mader, W.: Über die maximalzahl kantendisjunkter A-wege. Arch. Math. 30, 325–336 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  57. Mader, W.: Über die maximalzahl kreuzungsfreier H-wege. Arch. Math. 31, 387–402 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  58. Matula, D.: Determining edge-connectivity in O(mn). In: Proceeding of 28th Symp. Foundation Computer Science, 249–251 (1987)

    Google Scholar 

  59. Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math. 10, 96–115 (1927)

    MATH  Google Scholar 

  60. Nash-Williams, C.St.J.A.: Edge-disjonint spanning trees of finite graphs. J. Lond. Math. Soc. 36, 445–450 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  61. Nash-Williams, C.St.J.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39, 12 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  62. Oellermann, O.R.: Connectivity and edge-connectivity in graphs: A survey. Congessus Numerantium 116, 231–252 (1996)

    MathSciNet  MATH  Google Scholar 

  63. Oellermann, O.R.: On the ℓ-connectivity of a graph. Graphs Combin. 3 (1), 285–291 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  64. Oellermann, O.R.: A note on the ℓ-connectivity function of a graph. Congessus Numerantium 60, 181–188 (1987)

    MathSciNet  Google Scholar 

  65. Okamoto, F., Zhang, P.: The tree connectivity of regular complete bipartite graphs. J. Combin. Math. Combin. Comput. 74, 279–293 (2010)

    MathSciNet  MATH  Google Scholar 

  66. Ozeki, K., Yamashita, T.: Spanning trees: A survey. Graphs Combin. 27 (1), 1–26 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  67. Palmer, E.M.: On the spanning tree packing number of a graph: a survey. Discrete Math. 230 (1-3), 13–21 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  68. Peng, Y.H., Chen, C.C., Koh, K.M.: On the edge-toughness of a graph. Southeast Asian Bull. Math. 12 (2), 109–122 (1988)

    MathSciNet  MATH  Google Scholar 

  69. Petingi, L., Rodriguez, J.: Bounds on the maximum number of edge-disjoint Steiner trees of a graph. Congressus Numerantium 145, 43–52 (2000)

    MathSciNet  MATH  Google Scholar 

  70. Roskind, J., Tarjan, R.E.: A note on finding maximum-cost edge-disjoint spanning trees. Math. Oper. Res. 10 (4), 701–708 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  71. Schrijver, A.: Combinatorial optimization: Polyhedra and efficiency. Vol. B, Volume 24 of Algorithms and Combinatorics. Springer, Berlin (2003)

    Google Scholar 

  72. Sherwani, N.: Algorithms for VLSI Physical Design Automation, 3rd edn. Kluwer Academic. Pub., London (1999)

    MATH  Google Scholar 

  73. Sun, Y., Li, X.: On the difference of two generalized connectivities of a graph. J. Comb. Optim. (in press). DOI:10.1007/s10878-015-9956-9

    Google Scholar 

  74. Sun, Y., Zhou, S.: Tree connectivities of Caylay graphs on Abelian groups with small degrees. Bull. Malays. Math. Sci. Soc. (in press). DOI:10.1007/s40840-015-0147-8

    Google Scholar 

  75. Tutte, W.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 36 (1), 221–230 (1961)

    Article  MathSciNet  MATH  Google Scholar 

  76. Wang, H., Blough, D.M.: Construction of edge-disjoint spanning trees in the torus and application to multicast in wormhole-routed networks, In: Proc. Int’l Conf. Parallel and Distributed Computing Systems (1999)

    Google Scholar 

  77. Welsh, D.: Matroid Theorey. Academic Press, London (1976)

    Google Scholar 

  78. West, D.B., Wu, H.: Packing Steiner trees and S-connectors in graphs. J. Comb. Theory Ser. B 102 (1), 186–205 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  79. Whitney, H.: Congruent graphs and the connectivity of graph. Am. J. Math. 54 (1), 150–168 (1932)

    Article  MathSciNet  MATH  Google Scholar 

  80. Wilson, E.L., Hemminger, R.L., Plimmer, M.D.: A family of path properties for graphs. Math. Ann. 197 (2), 107–122 (1972)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2016 The Author(s)

About this chapter

Cite this chapter

Li, X., Mao, Y. (2016). Introduction. In: Generalized Connectivity of Graphs. SpringerBriefs in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-33828-6_1

Download citation

Publish with us

Policies and ethics