On the non-orientable genus of a 2-connected graph
In earlier works, additivity theorems for the genus and Euler genus of unions of graphs at two points have been given. In this work, the analogous result for the non-orientable genus is given. If \(\Sigma\) is obtained from the sphere by the addition of \(k>0\) crosscaps, define \(\tilde\gamma(\Sigma)\) to be k. For a graph G, define \({\tilde \gamma}\)(G) to be the least element in the set \(\{\tilde \gamma(\Sigma)|\) G embeds in \(\Sigma\}\). Theorem: Let \(H_ 1\) and \(H_ 2\) be connected graphs such that \(H_ 1\cap H_ 2\) consists of the isolated vertices v and w. Then, for some \(\mu \in \{-1,0,1,2\}\), \(\tilde \gamma(H_ 1\cup H_ 2) = \tilde\gamma(H_ 1)+\tilde\gamma(H_ 2)+\mu\). A formula for \(\mu\) is given.
scientific article; zbMATH DE number 3948296