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The number of edges that need to be removed from a graph to disconnect it is called the edge-connectivity. Similarly, given a graph of genus $n>0$, there is a minimum number of edges that you have to remove to obtain a graph of genus $n-1$. Is there a name for this number?

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Not as far as I know, and I've been doing this stuff for a long time. It's an interesting question, although I suspect that unless you put some more restrictions on the graphs to be considered you won't get any kind of nice answer. – Rick Decker Jun 11 '12 at 1:17
    
@RickDecker I was thinking that an alternative definition would focus on embeddings. Thus, given an embedding of genus $n$ (not necessarily a minimum genus embedding), I could ask for the minimum number of edges that need to be removed from the embedding to obtain an embedding of genus $n-1$. I am asking a related question here – becko Jun 11 '12 at 1:54

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