An edge subdivision is the insertion of a new vertex v_j in the middle of an exiting edge e = v_i v_k accompanied by the joining of the original edge endpoints with the new vertex to form new edges e' = v_i v_j and e'' = v_j v_k. A graph subdivision is therefore a sequence of edge subdivisions. Graphs for which there exists an isomorphism from a subdivision of one to a subdivision of the other are said to be homeomorphic graphs. In general, a graph simple unlabeled graph whose connectivity is considered purely on the basis of topological equivalence (i.e., up to smoothing and subdivision) is known as a topological graph. The opposite of graph subdivision is graph smoothing.
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