k-edge-connected graph

In graph theory, a connected graph is k-edge-connected if it remains connected whenever fewer than k edges are removed.

The edge-connectivity of a graph is the largest k for which the graph is k-edge-connected.

Edge connectivity and the enumeration of k-edge-connected graphs was studied by Camille Jordan in 1869.[1]

The edge connectivity version of Menger's theorem provides an alternative and equivalent characterization, in terms of edge-disjoint paths in the graph. If and only if every two vertices of G form the endpoints of k paths, no two of which share an edge with each other, then G is k-edge-connected. In one direction this is easy: if a system of paths like this exists, then every set X of fewer than k edges is disjoint from at least one of the paths, and the pair of vertices remains connected to each other even after X is deleted. In the other direction, the existence of a system of paths for each pair of vertices in a graph that cannot be disconnected by the removal of few edges can be proven using the max-flow min-cut theorem from the theory of network flows.

Edge connectivity is the dual concept to girth, the length of the shortest cycle in a graph, in the sense that the girth of a planar graph is the edge connectivity of its dual graph, and vice versa. These concepts are unified in matroid theory by the girth of a matroid, the size of the smallest dependent set in the matroid. For a graphic matroid, the matroid girth equals the girth of the underlying graph, while for a co-graphic matroid it equals the edge connectivity.[2]

The 2-edge-connected graphs can also be characterized by the absence of bridges, by the existence of an ear decomposition, or by Robbins' theorem according to which these are exactly the graphs that have a strong orientation.[3]

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).