Abstract
The problem of finding k-edge-connected components is a fundamental problem in computer science. Given a graph G = (V, E), the problem is to partition the vertex set V into {V 1, V 2,…, V h}, where each V i is maximized, such that for any two vertices x and y in V i, there are k edge-disjoint paths connecting them. In this paper, we present an algorithm to solve this problem for all k. The algorithm preprocesses the input graph to construct an Auxiliary Graph to store information concerning edge-connectivity among every vertex pair in O(Fn) time, where F is the time complexity to find the maximum flow between two vertices in graph G and n = ∣V∣. For any value of k, the k-edge-connected components can then be determined by traversing the auxiliary graph in O(n) time. The input graph can be a directed or undirected, simple graph or multigraph. Previous works on this problem mainly focus on fixed value of k.
Highlights
Graph connectivity is a fundamental problem in computer science, which has many background applications in the real world
The algorithm performs a preprocessing over the input graph to construct an auxiliary graph which is a tree in O(Fn) time, where F is the time complexity to find the maximum flow in the graph
Any improvement made on F automatically implies improvement to the time complexity of our algorithm
Summary
Graph connectivity is a fundamental problem in computer science, which has many background applications in the real world. Reliability in communication networks can be represented by the connectivity between each pair of nodes. In social networks, computing the closeness among people is a very important problem, which relates to the connectivity of the networks. There are many other applications which are related to the connectivity of networks, e.g., finding web pages of high commonality in internet searching; finding protein complexes and gene clusters in computational biology, etc. Graph connectivity has been well studied for more than forty years. It has a strong relationship with the problems of maximal network flow and minimal cut
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