# The 3-edge-components and a structural description of all 3-edge-cuts in a graph

## Abstract

Let *G = (V, E)* be an undirected graph, ¦*V*¦ = *n*. We denote *V*^{l} the partition of *V* into maximal vertex subsets indivisible by *k*(-edge)-cuts, *k < l*, of the whole *G*. The factor-graph of *G* corresponding to *V*^{3}, is known to give a clear representation of *V*^{2}, *V*^{3} and of the system of cuts of *G* with 1 and 2 edges. Here a (graph invariant) structural description of *V*^{4} and of the system of 3-cuts in an arbitrary graph *G* is suggested. It is based on a new concept of the 3-edge-connected components of a graph (with vertex sets from *V*^{3}). The 3-cuts of *G* are classified so that the classes are naturally 1∶1 correspondent to the 3-cuts of the 3-edge-connected components. A class can be reconstructed in a simple way from the component cut, using the relation of the component to the system of 2-cuts of *G*. For 3-cuts and *V*^{4} of a 3-edge-connected graph we follow [DKL76]. The space complexity of the description suggested is *O(n)* (though the total number of 3-cuts may be a cubic function of *n*).

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## References

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