In a graph G, contraction of an edge e with endpoints u, v is the replacement of u and v with a single vertex such that edges incident to the new vertex are the edges other than e that were incident with u or v. The resulting graph, denoted G·e, has one less edge than G. Edge contraction is implemented in the Wolfram Language as EdgeContract[g, e] or EdgeContract[g, elist]. Graph minors are defined in terms of edge contractions.

edge splitting | graph minor | vertex contraction