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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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    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.2012-07-18

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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.)2012-07-17