On the non-orientable genus of a 2-connected graph

AbstractIn 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 Σ is obtained from the sphere by the addition of k>0 crosscaps, define γ(Σ) to be k. For a graph G, define γ(G) to be the least element in the set {γ(Σ) | G embeds in Σ}.Theorem. Let H1 and H2 be connected graphs such that H1 ∩ H2 consists of the isolated vertices v and w. Then, for some μ ϵ −1, 0, 1, 2, γ(H1 ∪ H2) = γ(H1) + γ(H2) + μ.A formula for μ is given