On the Euler genus of a 2-connected graph (Q1074592)

The Euler genus of the surface \(\Sigma\) obtained from the sphere by the addition of k crosscaps and h handles in \(\epsilon (\Sigma)=k+2h\). For a graph G, the Euler genus \(\epsilon(G)\) of G is the smallest Euler genus among all surfaces in which G embeds. The following additivity theorem is proved: Suppose \(G=H\cup K\), where H and K have exactly the vertices v and w in common. Then \(\epsilon (G)=\min \{\epsilon (H+vw)+\epsilon (K+vw),\quad \epsilon (H)+\epsilon (K)+2\}.\)