Editing Homeomorphism (graph theory) (section)

===Reversion===
The reverse operation, '''smoothing out''' or '''smoothing'''  a vertex ''w'' with regards to the pair of edges (''e''<sub>1</sub>, ''e''<sub>2</sub>) incident on ''w'', removes both edges containing ''w'' and replaces (''e''<sub>1</sub>, ''e''<sub>2</sub>) with a new edge that connects the other endpoints of the pair. Here, it is emphasized that only [[degree (graph theory)|degree]]-2 (i.e., 2-valent) vertices can be smoothed. The limit of this operation is realized by the graph that has no  more [[degree (graph theory)|degree]]-2 vertices.

For example, the simple [[Connectivity (graph theory)|connected]] graph with two edges, ''e''<sub>1</sub> {''u'',''w''} and ''e''<sub>2</sub> {''w'',''v''}: 

[[Image:Graph subdivision step2.svg|150px|class=skin-invert]]

has a vertex (namely ''w'') that can be smoothed away, resulting in:

[[Image:Graph subdivision step1.svg|150px|class=skin-invert]]