Let's assume that I have a cubic 3-edge connected simple graph $G$. After taking a perfect matching (and we can specify which one we want), I want to split the remaining edges in 5 sets $U_1, ..., U_5$ such that all $G-U_i$ is 2-edge connected.
For example, if I have a Peterson graph, I can take a perfect matching (in orange).
Then, I can split the remaining edges in 5 sets (different colors).
Notice how if I omit the edges in one of the $U_i$ (color blue for the picture below), then the graph is 2-edge connected. By symmetry, this works for all the sets that I've chosen, even though this won't be the case for all graphs.
I've searched on the subject, but I've found nothing. I might just be searching for the wrong things though. Solving this would help with a problem I'm trying to find a good algorithm for. I need 5 different sets because I'm trying to achieve a certain ratio.
What I have found which was the closest is the following
Let G be a bridgeless cubic graph with specified edge e∗. Then there exists a 2-factor of G that covers all 3- and 4-edge cuts and does not contain e∗.
(T. Kaiser and R. Skrekovski ˇ , Cycles intersecting edge-cuts of prescribed sizes, SIAM J. Discrete Math., 22 (2008), pp. 861–874.)
While it may allow me to avoid the smaller cuts by requiring that all edges in a cut set not be of a same color, which would ensure 2-edge connectivity, I'm not sure how to show that.