Cycle separating cuts in possible counterexamples to the cycle double cover and the Berge-Fulkerson conjectures

It is known that smallest counterexamples to the Cycle Double Cover Conjecture and Berge-Fulkerson Conjecture (if they exist) are cyclically 4- and 5-edge-connected, respectively. We further analyse small cycle separating cuts in possible counterexamples. We prove that if a smallest counterexample G to the CDC Conjecture contains a cycle separating 4-cut S, then the behaviour of the admissible CDC coverings along the dangling edges of the two 4-poles induced by S is uniquely determined among more than 219 a priori possibilities. Similarly, for the Berge-Fulkerson Conjecture, we prove that among more than 2111 a priori possibilities, there are only 13 pairs of admissible sets that could occur along the dangling edges of a 5-cut in a smallest counterexample.