An edge subdivision is the insertion of a new vertex 
in the middle of an exiting edge 
accompanied by the joining of the original edge endpoints
with the new vertex to form new edges 
and 
(Gross and Yellen 2006, p. 293).
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.
