Is there a formal name for this graph operation? I'm writing a small function to alter a graph in a certain way and was wondering if there is a formal name for the operation.  The operation takes two distinct edges, injects a new node between the existing nodes of each edge and then adds an edge between the two new nodes.  For example:

add new nodes a and b to the graph
let edge1 = (x,y), let edge2 = (u,v)

delete edge (x,y)
create edges (x,a), (a,y)

delete edge(u,v)
create edges(u,b), (b,v)

create edge (a,b)
Your attempt is like subdivision. Actually you used two subdivision and finally you add extra edge between center of subdivision nodes.