TR#:  CS757 
Title:  THE 3EDGECOMPONENTS AND A STRUCTURAL DESCRIPTION OF ALL 3EDGECUTS IN A GRAPH. 
Authors:  E. Dinitz 
PDF  Revised  CS0757.revised.pdf 
Abstract:  Let G=(V,E) be an undirected graph, V=N. We denote {\Cal V}^L the partition of V into maximal vertex subsets indivisible by k(edge)cuts, k < l, of the whole G. The factorgraph of G corresponding to {\Cal V}^3, is known to give a clear representation of {\Cal V}^2, {\Cal V}^3 and of the system of cuts of G with 1 and 2 edges. Here a (graph invariant) structural description of {\Cal V}^4 and of the system of 3cuts in an arbitrary graph G is suggested. It is based on a new concept of the 3edgeconnected components of a graph (with vertex sets from {\Cal V}^3). The 3cuts of G are classified so that the classes are naturally 1:1 correspondent to the 3cuts of the 3edge 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 2cuts of G. For 3cuts and {\Cal V}^4 of a 3edgeconnected graph, we follow [DKL76]. The space complexity of the description suggested is O(n) (though the total number of 3cuts may be a cubic function of n).

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