K-edge-connected Graph - Computational Aspects

Computational Aspects

There is a polynomial-time algorithm to determine the largest k for which a graph G is k-edge-connected. A simple algorithm would, for every pair (u,v), determine the maximum flow from u to v with the capacity of all edges in G set to 1 for both directions. A graph is k-edge-connected if and only if the maximum flow from u to v is at least k for any pair (u,v), so k is the least u-v-flow among all (u,v).

If V is the number of vertices in the graph, this simple algorithm would perform iterations of the Maximum flow problem, which can be solved in time. Hence the complexity of the simple algorithm described above is in total.

An improved algorithm will solve the maximum flow problem for every pair (u,v) where u is arbitrarily fixed while v varies over all vertices. This reduces the complexity to and is sound since, if a cut of capacity less than k exists, it is bound to separate u from some other vertex.

A related problem: finding the minimum k-edge-connected subgraph of G (that is: select as few as possible edges in G that your selection is k-edge-connected) is NP-hard for .