Cut ideals of K4-minor free graphs are generated by quadrics

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 Petrović. We prove the full conjecture by introducing a new type of toric fiber product theorem.

[1]  Alexander Engström,et al.  Ideals of Graph Homomorphisms , 2010, 1002.4679.

[2]  Uwe Nagel,et al.  Properties of cut ideals associated to ring graphs , 2008, 0806.0585.

[3]  JOSEPH BRENNAN,et al.  Toric Geometry of Series-Parallel Graphs , 2009, SIAM J. Discret. Math..

[4]  S. Sullivant,et al.  Markov Bases of Binary Graph Models , 2003, math/0308280.

[5]  Seth Sullivant.  Toric fiber products , 2006 .

[6]  S. Sullivant,et al.  Toric geometry of cuts and splits , 2006, math/0606683.

[7]  Reinhard Diestel,et al.  Graph Theory , 1997 .