Generating Cut Conjunctions and Bridge Avoiding Extensions in Graphs

Let G = (V, E) be an undirected graph, and let B ⊆ V × V be a collection of vertex pairs. We give an incremental polynomial time algorithm to enumerate all minimal edge sets X ⊆ E such that every vertex pair (s, t) ∈ B is disconnected in (V, E r X), generalizing wellknown efficient algorithms for enumerating all minimal s-t cuts, for a given pair s, t ∈ V of vertices. We also present an incremental polynomial time algorithm for enumerating all minimal subsets X ⊆ E such that no (s, t) ∈ B is a bridge in (V, X ∪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 ∪ B, enumerate all minimal subsets X ⊆ E such that no element b ∈ B is spanned by ErX. Unlike the above special cases, corresponding to the cycle and cocycle matroids of the graph (V, E∪B), the enumeration of cut conjunctions for vectorial matroids turns out to be NP-hard.