Editing Homeomorphism (graph theory) (section)

==Embedding on a surface==
It is evident that subdividing a graph preserves [[planar graph|planarity]].  [[Kuratowski's theorem]] states that 
: a [[finite graph]] is planar [[if and only if]] it contains no [[Glossary of graph theory#Subgraphs|subgraph]] '''homeomorphic''' to ''K''<sub>5</sub> ([[complete graph]] on five vertices) or ''K''<sub>3,3</sub> ([[complete bipartite graph]] on six vertices, three of which connect to each of the other three).

In fact, a graph homeomorphic to ''K''<sub>5</sub> or ''K''<sub>3,3</sub> is called a [[Kuratowski's theorem|Kuratowski subgraph]].

A generalization, following from the [[Robertson–Seymour theorem]], asserts that for each [[integer]] ''g'', there is a finite '''obstruction set''' of graphs <math>L(g) = \left\{G_{i}^{(g)}\right\}</math> such that a graph ''H'' is [[graph embedding|embeddable]] on a surface of [[Genus (mathematics)|genus]] ''g'' if and only if ''H'' contains no homeomorphic copy of any of the <math>G_{i}^{(g)\!}</math>.  For example, <math>L(0) = \left\{K_{5}, K_{3,3}\right\}</math> consists of the Kuratowski subgraphs.