In combinatorics there are very simple basic graphs from which a whole lot of theory came. For example the complete graphs `K_5` and `K_{3,3}` which alone provide the ground level for any non-planar graph according to Kuratowski's theorem. Another simple graph that gave rise to a huge amount of theory is Petersen's graph, which I like to think as the graph whose vertices are the ten two-element subsets of `{1,2,3,4,5}`, and for which two such vertices are connected iff they are disjoint.

A link for Kuratowski's theorem is [http://en.wikipedia.org/wiki/Kuratowski's_theorem][1]


  [1]: http://en.wikipedia.org/wiki/Kuratowski's_theorem