# subdivision

A graph $H$ is said to be a subdivision, or topological minor of a graph $G$, or a topological $G$ graph if $H$ is obtained from $G$ by subdividing some of the edges, that is, by replacing the edges by paths having at most their endvertices in common. We often use $TG$ for a topological $G$ graph.

Thus, $TG$ denotes any member of a large family of graphs; for example, $TC_{4}$ is an arbitrary cycle of length at least 4. For any graph $G$, the spaces $R(G)$ (denoting the realization of G) and $R(TG)$ are homeomorphic.

Adapted with permission of the author from by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title subdivision Subdivision 2013-03-22 12:31:51 2013-03-22 12:31:51 CWoo (3771) CWoo (3771) 5 CWoo (3771) Definition msc 05C99 topological minor Homeomorphic Realization