Suppose a graph $G$ has edge-connectivity $c$. The min-cuts of $G$ (the cuts of weight $c$) can be represented in terms of a cactus graph $H$. This is "well-known".
Each vertex of $w \in H$ corresponds to a subset of vertices $v_1, \dots, v_k \in G$. Do the vertices $v_1, \dots, v_k$ need to be connected in $G$? Must they be highly-connected, such as $c/2$-connected or something like this? Any pointers to a good reference about the cactus graph and its properties would be very helpful too.
Thanks for the help!