On the Euler genus of a 2-connected graph

AbstractThe Euler genus of the surface Σ obtained from the sphere by the addition of k crosscaps and h handles is ε(Σ) = k + 2h. For a graph G, the Euler genus ε(G) of G is the smallest Euler genus among all surfaces in which G embeds. The following additivity theorem is proved.Theorem. Suppose G = H ∪ K, where H and K have exactly the vertices v and win common. Then ε(G) = min(ε(H + vw) + ε(K + vw), ε(H) + ε(K) + 2)