Abstract
A classical enumerative result states that, given a graph G and a vertex u, the number of connected subgraphs of G is equal to the number of orientations of G such that every vertex can reach u by a directed path. We show that this result is an instance of a much broader set of enumerative identities between subgraphs and orientations corresponding to various connectivity constraints. Namely, given two sets of pairs of vertices A = {(u i , v i), i ∈ [k]} and B = {(u ′ i , v ′ i), i ∈ [ℓ]}, we consider the orientations α of G such that adding the elements of A and B as additional directed edges to α gives an orientation α ′ in which v i cannot reach u i for all i ∈ [k], but v ′ i can reach u ′ i for all i ∈ [ℓ]. We show that this set of orientations is equinumerous to a set of subgraphs satisfying the "same" connectivity constraints defined in terms of A and B. We also extend our results to the enumeration of equivalence classes of orientations satisfying such connectivity constraints. Precisely, we consider the equivalence classes under cycle reversal, cocycle reversal, or cycle-cocycle reversal. We show that the equivalences classes are equinumerous to some sets of subgraphs defined by connectivity and acyclicity constraints.