Let $G=(V,E)$ be an undirected graph, and let $B \subseteq
V \times V$ be a collection of vertex pairs. We give an incremental
polynomial time algorithm to enumerate all minimal edge sets $X
\subseteq E$ such that every vertex pair $(s,t) \in B$ is
disconnected in $(V,E \smallsetminus X)$, generalizing well-known
efficient algorithms for enumerating all minimal $s$-$t$ cuts, for a
given pair $s,t \in V$ of vertices. We also present an incremental
polynomial time algorithm for enumerating all minimal subsets $X
\subseteq E$ such that no $(s,t) \in B$ is a bridge in $(V,X \cup
B)$. These two enumeration problems are special cases of the more
general cut conjunction problem in matroids: given a matroid $M$ on
ground set $S=E \cup B$, enumerate all minimal subsets $X \subseteq
E$ such that no element $b \in B$ is spanned by $E \smallsetminus
X$. Unlike the above special cases, corresponding to the cycle and
cocycle matroids of the graph $(V,E \cup B)$, the enumeration of cut
conjunctions for  vectorial matroids turns out to be NP-hard.