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It is known that the orientable genus of union of two (disjoint) graphs is the sum of their genus. So, it is natural to ask

What can be said about the non-orientable genus of union of two (disjoint) graphs?

Note that upper and lower bounds are known for the non-orientable genus of $k$-amalgams of two graphs with $k\geq2$ as well as the precise value of the non-orientable genus for $k=1$.

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  • $\begingroup$ Stahl and Beineke claim in the last paragraph of their paper: "We note here that Corollary 3 retains its validity when the phrase blocks of the connected graph is replaced by components of the graph." I don't see why that's true though. $\endgroup$
    – Jan Kyncl
    Oct 26, 2017 at 15:07

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The non-orientable genus $\tilde{\rm{g}}$ is not additive: consider the union of $K_5$ and $K_7$. We have $\tilde{\rm{g}}(K_5)=1$, $\tilde{\rm{g}}(K_7)=3$ and $\tilde{\rm{g}}(K_5\cup K_7)=3$: embed $K_5$ on the projective plane, $K_7$ on the torus, and take the connected sum of these two surfaces.

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