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Is there a special name for the kind of subgraphs you get by taking some sequence of the following operation: Pick two vertices and identify them so all edges going to either vertex get sent to the new vertex.

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Note that if you contract only adjacent vertices of a graph $G$ you obtain a minor of $G$. –  Jernej Jul 17 '12 at 6:48
Here is link to Wikipedia article on minors mentioned in Azoo's comment. –  Martin Sleziak Jul 17 '12 at 12:28
I'm actually doing the opposite here and only contracting vertices which are not adjacent. My graph is the Hasse graph of a ranked poset and my identifications take place between vertices of the same rank. –  user1390 Jul 18 '12 at 3:52

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You don't get a subgraph when you do that, but what you might call a quotient graph instead (the natural map goes the other way). The operation is called vertex contraction.

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I like the name for this as it goes along with identification similar to the quotient topology. Thank you! (I was aware the output would not necessarily be a subgraph cause it's easy to see the maximal and average degree can increase, but I poorly phrased my question and used subgraph there instead of output.) –  user1390 Jul 17 '12 at 13:32

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