Given a finite undirected graph G and A ⊆ E(G), G(A) denotes the subgraph of G having edge-set A and having no isolated vertices. For a partition {E1, E2} of E(G), W(G; E1) denotes the set V(G(E1)) ⋂ V(G(E2)). We say that Gis non-separable if it is connected and for every proper, non-empty subset A of E(G), we have |W(G; A)| ≧ 2. A split of a non-separable graph Gis a partition {E1, E2} of E(G) such that
|E1| ≧ 2 ≧ |E2| and |W(G; E1)| = 2.
Where {E1, E2} is a split of G, W(G; E2) = {u, v}, and e is an element not in E(G), we form graphs Gii= 1 and 2, by adding e to G(Ei) as an edge joining u to v.