Graph smoothing, also known as smoothing away or smoothing out, is the process of replacing edges 
and 
incident at a vertex 
of vertex degree 2 by a single new edge 
and removing the vertex 
(Gross and Yellen 2006, p. 293).
A tree which is smoothed until no vertices of degree two remain is known as a series-reduced tree. 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 process of smoothing simpe cyclic graphs is less well defined, since while a single smoothing of the cycle graph 
gives the graph 
for 
, if additional smoothing is performed, the graph 
is smoothed to the dipole
graph 
which is no longer a simple graph but rather a multigraph
since it contains two edges between its two vertices. Similarly, smoothing 
give the bouquet graph 
which is no longer a simple graph
but rather a pseudograph since it consists of a single
vertex connected to itself by a graph loop. Finally,
according to Gross and Yellen (2006, p. 293), it is not permitted to smooth
away the sole remaining vertex of 
.
Graph smoothing is the opposite of graph subdivision.
