Graph Subdivision
An edge subdivision is the insertion of a new vertex v_j in the middle of an exiting edge e=v_iv_k accompanied by the joining of the original edge endpoints with the new vertex to form new edges e^'=v_iv_j and e^('')=v_jv_k (Gross and Yellen 2006, p. 293).
A graph subdivision is therefore a sequence of edge subdivisions.
Graphs for which there exists an isomorphism from a subdivision of one to a subdivision of the other are said to be homeomorphic graphs.
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 opposite of graph subdivision is graph smoothing.
The subdivided cubical graph is its own graph distance-5 graph and the the subdivided dodecahedral graph is its own graph distance-9 graph.
See also
Graph Smoothing, Homeomorphic Graphs, Topological GraphExplore with Wolfram|Alpha
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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.Referenced on Wolfram|Alpha
Graph SubdivisionCite this as:
Weisstein, Eric W. "Graph Subdivision." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/GraphSubdivision.html