  Cut ideals are used in algebraic statistics to study statistical models
defined by graphs. Intuitively, topological restrictions on the graphs should
imply structural statements about the corresponding cut ideals. Several
theorems and many computer calculations support that.
  Sturmfels and Sullivant conjectured that the cut ideal is generated by
quadrics if and only if the graph is free of K4-minors. Parts of the conjecture
has been resolved by Brennan and Chen, and later by Nagel and Petrovic.
  We prove the full conjecture by introducing a new type of toric fiber product
theorem.
