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The 3-edge-components and a structural description of all 3-edge-cuts in a graph. (English) Zbl 0789.05077
Mayr, Ernst W. (ed.), Graph-theoretic concepts in computer science. 18th international workshop, WG ’92, Wiesbaden-Naurod, Germany, June 18-20, 1992. Proceedings. Berlin: Springer-Verlag. Lect. Notes Comput. Sci. 657, 145-157 (1993).
Summary: Let \(G=(V,E)\) be an undirected graph, \(| V |=n\). We denote \({\mathcal V}^ l\) 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 \({\mathcal V}^ 3\), is known to give a clear representation of \({\mathcal V}^ 2\), \({\mathcal V}^ 3\) and of the system of cuts of \(G\) with 1 and 2 edges. Here a (graph invariant) structural description of \({\mathcal V}^ 4\) 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 \({\mathcal V}^ 3)\). 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 \(G\). The space complexity of the description suggested is \(O(n)\) (though the total number of 3-cuts may be a cubic function of \(n)\).
For the entire collection see [Zbl 0866.00058].
MSC:
05C70 Edge subsets with special properties (factorization, matching, partitioning, covering and packing, etc.)
05C75 Structural characterization of families of graphs
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