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 V1 the partition of V into maximal vertex subsets indivisible by k(-edge)-cuts1 k < l, of the whole G. The factor-graph of G corresponding to V3 is known to give a clear representation of V2, V3 and of the system of cuts of G with 1 and 2 edges. Here a (graph invariant) structural description of V4 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 V3). 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 V4 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).