Maintaining the 4-edge-connected components of a graph on-line

Two vertices v and u of an undirected graph are called k-edge-connected if there exist k edge-disjoint paths between v and u. The equivalence classes of this relation are called the k-edge-connected components. The author suggests graph structures and an incremental algorithm to maintain k-edge-connected components for the case k=4. Any sequence of a q queries Same-k-Component? and updates Insert-Edge on an n-vertex graph can be performed in O(q sigma (q,n)+n log n) time, with O(m+n log n) preprocessing (m is the number of edges in the initial graph). Besides, an algorithm for maintaining k-edge-connected components (k arbitrary) in a (k-1)-edge-connected graph is presented. The complexity is O((q+n) alpha (q,n)), with O(m+k/sup 2/n log(n/k)) preprocessing.<<ETX>>

[1]  Harold N. Gabow,et al.  Applications of a poset representation to edge connectivity and graph rigidity , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[2]  Zvi Galil,et al.  Maintaining the 3-Edge-Connected Components of a Graph On-Line , 1993, SIAM J. Comput..

[3]  R. Bixby.  The Minimum Number of Edges and Vertices in a Graph with Edge Connectivity n and m n-Bonds , 1974, Networks.

[4]  Jan van Leeuwen,et al.  Worst-case Analysis of Set Union Algorithms , 1984, JACM.

[5]  Roberto Tamassia,et al.  On-Line Graph Algorithms with SPQR-Trees , 1990, ICALP.

[6]  David Eppstein,et al.  Sparsification-a technique for speeding up dynamic graph algorithms , 1992, Proceedings., 33rd Annual Symposium on Foundations of Computer Science.

[7]  Roberto Tamassia,et al.  On-line maintenance of the four-connected components of a graph , 1991, [1991] Proceedings 32nd Annual Symposium of Foundations of Computer Science.

[8]  Yefim Dinitz.  The 3-Edge-Components and a Structural Description of All 3-Edge-Cuts in a Graph , 1992, WG.

[9]  Zvi Galil,et al.  Reducing edge connectivity to vertex connectivity , 1991, SIGA.