Incremental Maintenance of the 5-Edge-Connectivity Classes of a Graph

Authors:
Yefim Dinitz;Ronit Nossenson
Affiliations:
-;-
Venue:
SWAT '00 Proceedings of the 7th Scandinavian Workshop on Algorithm Theory
Year:
2000

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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 ċ log2n); the worst-case time per query is O(1).