Abstract
Two vertices of an undirected graph are called k-edge-connected if there exist k edge-disjoint paths between them. The equivalence classes of this relation are called k-edge-connected classes, or k-classes for short. This paper shows how to check whether two vertices belong to the same 5-class of an arbitrary connected graph that is undergoing edge insertions. For this purpose we suggest (i) a full description of the 4-cuts of an arbitrary graph and (ii) a representation of the k-classes, 1 = k = 5, of size linear in n-the number of vertices of the graph; these representations can be constructed in a polynomial time. Using them, we suggest an algorithm for incremental maintenance of the 5-classes. The total time for a sequence of m Edge-Insert updates and q Same-5-Class? queries is O(q + m + n · log2n); the worst-case time per query is O(1).
Keywords
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.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
A. A. Benczur, “Augmenting undirected connectivity in O(n3) time”, Proc. 26th Annual ACM Symp. on Theory of Computing, ACM Press, 1994, 658–667.
A. A. Benczur, “The structure of near-minimum edge cuts”, In Proc. 36st Annual Symp. on Foundations of Computer Science, 1995, 92–102.
T. Cormen, C. Leiserson, and R. Rivest. Introduction to Algorithms, McGraw-Hill, New York, NY, 1990.
Ye. Dinitz, “The 3-edge components and the structural description of all 3-edge cuts in a graph”, Proc. 18th International Workshop on Graph-Theoretic Concepts in Computer Science (WG92), Lecture Notes in Computer Science, v.657, Springer-Verlag, 1993, 145–157.
Ye. Dinitz, “The 3-edge components and the structural description of edge cuts in a graph”, Manuscript.
E. A. Dinic, A. V. Karzanov and M. V. Lomonosov, “On the structure of the system of minimum edge cuts in a graph”, Studies in Discrete Optimization, A. A. Fridman (Ed.), Nauka, Moscow, 1976, 290–306 (in Russian).
Ye. Dinitz and Z. Nutov, “A 2-level cactus tree model for the minimum and minimum+1 edge cuts in a graph and its incremental maintenance”, Proc. the 27th Symposium on Theory of Computing, 1995, 509–518.
Ye. Dinitz and A. Vainshtein, “The connectivity carcass of a vertex subset in a graph and its incremental maintenance”, Proc. 26th Annual ACM Symp. on Theory of Computing, ACM Press, 1994, 716–725 (see also TR-CS0804 and TR-CS0921, Technion, Haifa, Israel).
Ye. Dinitz and J. Westbrook, “Maintaining the Classes of 4-Edge-Connectivity in a Graph On-Line”, Algorithmica 20 (dy1998), no. 3, 242–276.
H. N. Gabow, “Applications of a poset representation to edge connectivity and graph rigidity”, Proc. 23rd Annual ACM Symp. on Theory of Computing, 1991, 112–122.
R. E. Gomory and C. T. Hu, “Multi-terminal network flows”, J. SIAM 9(4) (dy1961), 551–570.
Z. Galil and G. F. Italiano, “Maintaining the 3-edge-connected components of a graph on line”, SIAM J. Computing 22(1), 1993, 11–28.
F. Harary. Graph Theory, Addison-Wesley, Reading, MA, 1972.
Hopcroft, J., and Tarjan, R.E., Dividing a graph into triconnected components. SIAM J. Comput. 2 (dy1973) 135–158.
J. A. La Poutré, J. van Leeuwen, and M. H. Overmars. Maintenance of 2-and 3-edge-connected components of graphs. Discrete Mathematics 114, 1993, 329–359.
D. Naor, D. Guisfield and C. Martel, “A fast algorithm for optimally increasing the edge connectivity”, In Proc. 31st Annual Symp. on Foundations of Computer Science, 1990, 698–707.
D. D. Sleator and R. E. Tarjan, “A data structure for the dynamic trees”, In Proc. 13th Annual ACM Symposium on Theory of Computing, 1981, 114–122.
Tutte, W.T., Connectivity in Graphs, Univ. of Toronto Press, Toronto, 1966.
Teplixke, R., Dynamic Maintenance of Connectivity Classes of a Graph, Using Decomposition into 3-Components, M. Sc. Thesis, the Technion, Haifa, Israel, 1999.
J. Westbrook and R. E. Tarjan, “Maintaining bridge-connected and biconnected components on line”, Algorithmica, 7 (dy1992), 433–464.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2000 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Dinitz, Y., Nossenson, R. (2000). Incremental Maintenance of the 5-Edge-Connectivity Classes of a Graph. In: Algorithm Theory - SWAT 2000. SWAT 2000. Lecture Notes in Computer Science, vol 1851. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-44985-X_25
Download citation
DOI: https://doi.org/10.1007/3-540-44985-X_25
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-67690-4
Online ISBN: 978-3-540-44985-0
eBook Packages: Springer Book Archive