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).
This research was done partly when the author was with IBM Israel, Science & Technology, Haifa.
In previous papers: E.A.Dinic, Moscow.
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References
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Dinitz, E. (1993). The 3-edge-components and a structural description of all 3-edge-cuts in a graph. In: Mayr, E.W. (eds) Graph-Theoretic Concepts in Computer Science. WG 1992. Lecture Notes in Computer Science, vol 657. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-56402-0_44
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DOI: https://doi.org/10.1007/3-540-56402-0_44
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