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 .