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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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    @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](http://math.stackexchange.com/q/156182/10063)2012-06-11

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