Graph Smoothing
المؤلف:
Gross, J. T. and Yellen, J.
المصدر:
Graph Theory and Its Applications, 2nd ed. Boca Raton, FL: CRC Press
الجزء والصفحة:
p. 293
1-5-2022
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Graph Smoothing
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.
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.
REFERENCES
Gross, J. T. and Yellen, J. Graph Theory and Its Applications, 2nd ed. Boca Raton, FL: CRC Press, p. 293, 2006.
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