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Graph Smoothing


GraphSmoothing

Graph smoothing, also known as smoothing away or smoothing out, is the process of replacing edges e^'=v_iv_j and e^('')=v_jv_k incident at a vertex v_j of vertex degree 2 by a single new edge e=v_iv_k and removing the vertex v_j (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 C_n gives the graph C_(n-1) for n>3, if additional smoothing is performed, the graph C_3 is smoothed to the dipole graph D_2 which is no longer a simple graph but rather a multigraph since it contains two edges between its two vertices. Similarly, smoothing D_2 give the bouquet graph B_1 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 B_1.

Graph smoothing is the opposite of graph subdivision.


See also

Graph Subdivision, Homeomorphic Graphs, Series-Reduced Tree, Topological Graph

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References

Gross, J. T. and Yellen, J. Graph Theory and Its Applications, 2nd ed. Boca Raton, FL: CRC Press, p. 293, 2006.

Cite this as:

Weisstein, Eric W. "Graph Smoothing." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/GraphSmoothing.html

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