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I'm trying to describe a sort of join between two graphs $G$ and $H$ where you delete one edge from each graph (let's call the vertices adjacent to the deleted edges $g_1$, $g_2$, $h_1$ and $h_2$) , and then join the two graphs by adding an edge between $g_1$ and $h_1$, and another between $g_2$ and $h_2$.

Does this join have a special name? I can't seem to find one, although I have a feeling it has one.

Thanks!

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The operation of deleting a pair of parallel edges and replacing them with a disjoint pair arises in the study of graphs with given degree sequence, but even there I do not recall it having a name. (Swapping 2-matchings?) –  Chris Godsil Feb 4 '13 at 12:31
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Unfortunately I'm not sure if this has a name in graph theory, but it reminds me a lot of the knot sum in knot theory.

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