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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Bibliography
Barden, B., Libeskind-Hadas, R., Davis, J., Williams, W.: On edge-disjoint spanning trees in hypercubes. Inform. Proc. Lett. 70 (1), 13–16 (1999)
Bauer, D., Broersma, H., Schmeichel, E.: Toughness in graphs: A survey. Graphs Combin. 22, 1–35 (2006)
Beineke, L.W., Wilson, R.J.: Topics in Structural Graph Theory. Cambrige University Press, Cambrige (2013)
Beineke, L.W., Oellermann, O.R., Pippert, R.E.: The average connectivity of a graph. Discrete Math. 252 (1-3), 31–45 (2002)
Boesch, F.T., Chen, S.: A generalization of line connectivity and optimally invulnerable graphs. SIAM J. Appl. Math. 34 (4), 657–665 (1978)
Bollobás, B.: Extremal Graph Theory. Academic press, New York (1978)
Bondy, J.A., Murty, U.S.R.: Graph Theory with Applications. North-Holland, Amsterdam (1976)
Bondy, J.A., Murty, U.S.R.: Graph Theory, GTM 244. Springer, New York (2008)
Cai, Q., Li, X., Song, J.: Solutions to conjectures on the (k, ℓ)-rainbow index of complete graphs. Networks 62, 220–224 (2013)
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)
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)
Chartrand, G., Kapoor, S.F., Lesniak, L., Lick, D.R.: Generalized connectivity in graphs. Bull. Bombay Math. Colloq. 2, 1–6 (1984)
Chartrand, G., Okamoto, F., Zhang, P.: Rainbow trees in graphs and generalized connectivity. Networks 55 (4), 360–367 (2010)
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
Chen, X., Li, X., Lian, H.: Note on packing of edge-disjoint spanning trees in sparse random graphs. arXiv:1301.1097 [math.CO] (2013)
Chen, L., Li, X., Yang, K., Zhao, Y.: The 3-rainbow index of a graph. Discuss. Math. Graph Theory 35 (1), 81–94 (2015)
Cheng, X., Du, D.: Steiner Trees in Industry. Kluwer Academic Publisher, Dordrecht (2001)
Chvátal, V.: Tough graphs and hamiltonian circuits. Discrete Math. 5 (3), 215–228 (1973)
Cunningham, W.H.: Optimal attack and reinforcement of a netwok. J. ACM 32 (3), 549–561 (1985)
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)
Dirac, G.A.: In abstrakten graphen vorhandene vollständige 4-graphen und ihre unterteilungen. Math. Nach 22, 61–85 (1960)
Du, D., Hu, X.: Steiner Tree Problems in Computer Communication Networks. World Scientific, Singapore (2008)
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)
Esfahanian, A.H.: Generalized measures of fault tolerance with application to N-cube networks. IEEE Trans. Comput. 38, 1586–1591 (1989)
Ford, L.R., Fulkerson, D.R.: Maximum flow through a network. Can. J. Math. 8, 399–404 (1956)
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)
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)
Goldsmith, D.L.: On the second-order edge-connectivity of a graph. Congressus Numerantium 29, 479–484 (1980)
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)
Grötschel, M.: The Steiner tree packing problem in VLSI design. Math. Program. 78, 265–281 (1997)
Grötschel, M., Martin, A., Weismantel, R.: Packing Steiner trees: A cutting plane algorithm and commputational results. Math. Program. 72, 125–145 (1996)
Gu, R., Li, X., Shi, Y.: The generalized 3-connectivity of random graphs. Acta Math. Sin. (Chinese Edition) 57 (2), 321–330 (2014)
Gusfield, D.: Connectivity and edge-disjoint spanning trees. Infor. Process. Lett. 16 (2), 87–89 (1983)
Hager, M.: Pendant tree-connectivity. J. Comb. Theory 38 (2), 179–189 (1985)
Hager, M.: Path-connectivity in graphs. Discrete Math. 59 (1-2), 53–59 (1986)
Harary, F.: Graph Theory. Addison-Wesley, Reading (1969)
Itai, A., Rodeh, M.: The multi-tree approach to reliability in distributed networks. Inform. Comput. 79 (1), 43–59 (1988)
Jain, K., Mahdian, M., Salavatipour, M.: Packing Steiner trees. In: Proc. 14th ACM-SIAM symposium on Discterte Algorithms, Baltimore, pp. 266–274 (2003)
Kriesell, M.: Edge-disjoint trees containing some given vertices in a graph. J. Combin. Theory Ser. B 88 (1), 53–65 (2003)
Kriesell, M.: Local spanning trees in graphs and hypergraph decomposition with respect to edge-connectivity. Electron. Notes Discrete Math. 3, 110–113 (1999)
Lau, L.: An approximate max-Steiner-tree-packing min-Steiner-cut theorem. Combinatorica 27, 71–90 (2007)
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)
Li, X., Mao, Y.: Nordhaus-Gaddum-type results for the generalized edge-connectivity of graphs. Discrete Appl. Math. 185, 102–112 (2015)
Li, X., Mao, Y.: The generalized 3-connectivity of lexicographic product graphs. Discrete Math. Theor. Comput. Sci. 16 (1), 339–354 (2014)
Li, X., Mao, Y.: The minimal size of a graph with given generalized 3-edge-connectivity. Ars Combin. 118, 63–72 (2015)
Li, X., Mao, Y., Sun, Y.: On the generalized (edge-)connectivity of graphs. Australasian J. Combin. 58 (2), 304–319 (2014)
Li, X., Mao, Y.: Graphs with large generalized (edge-)connectivity. Discuss. Math. Graph Theory. Accepted by arXiv: 1305.1089 [math.CO] (2013)
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)
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)
Li, X., Shi, Y., Sun, Y.: Rainbow connections of graphs: A survey. Graphs Combin. 29 (1), 1–38 (2013)
Li, X., Sun, Y.: Rainbow Connections of Graphs. Springer Briefs in Math. Springer, New York (2012)
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)
Li, X., Zhao, Y.: On graphs with only one Steiner tree connecting any k vertices. arXiv:1301.4623v1 [math.CO] (2013)
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)
Lü, M., Chen, G.L., Xu, J.: On super edge-connectivity of Cartesian product graphs. Networks 49 (2), 135–157 (2007)
Mader, W.: Über die maximalzahl kantendisjunkter A-wege. Arch. Math. 30, 325–336 (1978)
Mader, W.: Über die maximalzahl kreuzungsfreier H-wege. Arch. Math. 31, 387–402 (1978)
Matula, D.: Determining edge-connectivity in O(mn). In: Proceeding of 28th Symp. Foundation Computer Science, 249–251 (1987)
Menger, K.: Zur allgemeinen Kurventheorie. Fund. Math. 10, 96–115 (1927)
Nash-Williams, C.St.J.A.: Edge-disjonint spanning trees of finite graphs. J. Lond. Math. Soc. 36, 445–450 (1961)
Nash-Williams, C.St.J.A.: Decomposition of finite graphs into forests. J. Lond. Math. Soc. 39, 12 (1964)
Oellermann, O.R.: Connectivity and edge-connectivity in graphs: A survey. Congessus Numerantium 116, 231–252 (1996)
Oellermann, O.R.: On the ℓ-connectivity of a graph. Graphs Combin. 3 (1), 285–291 (1987)
Oellermann, O.R.: A note on the ℓ-connectivity function of a graph. Congessus Numerantium 60, 181–188 (1987)
Okamoto, F., Zhang, P.: The tree connectivity of regular complete bipartite graphs. J. Combin. Math. Combin. Comput. 74, 279–293 (2010)
Ozeki, K., Yamashita, T.: Spanning trees: A survey. Graphs Combin. 27 (1), 1–26 (2011)
Palmer, E.M.: On the spanning tree packing number of a graph: a survey. Discrete Math. 230 (1-3), 13–21 (2001)
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)
Petingi, L., Rodriguez, J.: Bounds on the maximum number of edge-disjoint Steiner trees of a graph. Congressus Numerantium 145, 43–52 (2000)
Roskind, J., Tarjan, R.E.: A note on finding maximum-cost edge-disjoint spanning trees. Math. Oper. Res. 10 (4), 701–708 (1985)
Schrijver, A.: Combinatorial optimization: Polyhedra and efficiency. Vol. B, Volume 24 of Algorithms and Combinatorics. Springer, Berlin (2003)
Sherwani, N.: Algorithms for VLSI Physical Design Automation, 3rd edn. Kluwer Academic. Pub., London (1999)
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
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
Tutte, W.: On the problem of decomposing a graph into n connected factors. J. Lond. Math. Soc. 36 (1), 221–230 (1961)
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)
Welsh, D.: Matroid Theorey. Academic Press, London (1976)
West, D.B., Wu, H.: Packing Steiner trees and S-connectors in graphs. J. Comb. Theory Ser. B 102 (1), 186–205 (2012)
Whitney, H.: Congruent graphs and the connectivity of graph. Am. J. Math. 54 (1), 150–168 (1932)
Wilson, E.L., Hemminger, R.L., Plimmer, M.D.: A family of path properties for graphs. Math. Ann. 197 (2), 107–122 (1972)
Author information
Authors and Affiliations
Rights 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
DOI: https://doi.org/10.1007/978-3-319-33828-6_1
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-33827-9
Online ISBN: 978-3-319-33828-6
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)