Joins from cuts Throughout this section we assume that we are given two labeled, connected, undirected graphs G = (V, E) and H = (W, F ) as well as proper subsets ∅ ⊊ V ′ ⊊ V and ∅ ⊊ W ′ ⊊ W of their vertex sets. Writing V ′′ := V \ V ′ and W ′′ := W \ W ′ , moreover it seems natural to associate the newly inserted edges with the edge cuts