Geodesic
On a~decomposition of a~$3$-connected graph into cyclically $4$-edge-connected components
Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VIII, Tome 450 (2016), pp. 109-150

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A graph is called cyclically $4$-edge-connected if removing any three edges from it leads us to a graph, at most one connected component of which contains a cycle. $3$-connected graph is $4$-edge-connected iff removing any three edges from it leads us to either a connected graph or to a graph with exactly two connected components, one of which is a single-vertex one. We show, how to correspond for any $3$-connected graph a components tree, such that every component would be a $3$-connected and cyclically $4$-edge-connected graph. 
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     title = {On a~decomposition of a~$3$-connected graph into cyclically $4$-edge-connected components},
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A. V. Pastor. On a~decomposition of a~$3$-connected graph into cyclically $4$-edge-connected components. Zapiski Nauchnykh Seminarov POMI, Combinatorics and graph theory. Part VIII, Tome 450 (2016), pp. 109-150. http://geodesic.mathdoc.fr/item/ZNSL_2016_450_a6/