A graph is called cyclically 4-edge-connected if removing any three edges from it results in a graph in which at most one connected component contains a cycle. A 3-connected graph is 4-edge-connected if and only if removing any three edges from it results in either a connected graph or a graph with exactly two connected components one of which is a single-vertex one. We show how to associate with any 3-connected graph a tree of components such that every component is a 3-connected and cyclically 4-edge-connected graph.
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D. V. Karpov, “Blocks in k-connected graphs,” Zap. Nauchn. Semin. POMI, 293, 59–93 (2002).
D. V. Karpov, “Cutsets in a k-connected graph,” Zap. Nauchn. Semin. POMI, 340, 33–60 (2006).
D. V. Karpov, “The decomposition tree of a biconnected graph,” Zap. Nauchn. Semin. POMI, 417, 86–105 (2013).
D. V. Karpov, “The tree of cuts and minimal k-connected graphs,” Zap. Nauchn. Semin. POMI, 427, 22–40 (2014).
D. V. Karpov and A. V. Pastor, “On the structure of a k-connected graph,” Zap. Nauchn. Semin. POMI, 266, 76–106 (2000).
D. V. Karpov and A. V. Pastor, “The structure of a decomposition of a triconnected graph,” Zap. Nauchn. Semin. POMI, 391, 90–148 (2011).
F. Harary, Graph Theory, Addison-Wesley (1969).
W. Hohberg, “The decomposition of graphs into k-connected components,” Discrete Math., 109, 133–145 (1992).
W. T. Tutte, Connectivity in Graphs, Univ. Toronto Press, Toronto (1966).
Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 450, 2016, pp. 109–150.
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Pastor, A.V. On the Decomposition of a 3-Connected Graph into Cyclically 4-Edge-Connected Components. J Math Sci 232, 61–83 (2018). https://doi.org/10.1007/s10958-018-3859-0