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.

Let ${\displaystyle G=(V,E)}$ be an arbitrary graph. If subgraph ${\displaystyle G'=(V,E\setminus X)}$ is connected for all ${\displaystyle X\subseteq E}$ where ${\displaystyle |X|<k}$, then G is k-edge-connected. The edge connectivity of ${\displaystyle G}$ is the maximum value k such that G is k-edge-connected. The smallest set X whose removal disconnects G is a minimum cut in G.

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