This paper studies a class of binomial ideals associated to graphs with finite vertex sets. They generalize the binomial edge ideals, and they arise in the study of conditional independence ideals. A Gröbner basis can be computed by studying paths in the graph. Since these Gröbner bases are square-free, generalized binomial edge ideals are radical. To find the primary decomposition a combinatorial problem involving the connected components of subgraphs has to be solved. The irreducible components of the solution variety are all rational. © 2012 Elsevier Inc.
Rauh, J. (2013). Generalized binomial edge ideals. Advances in Applied Mathematics, 50(3), 409–414. https://doi.org/10.1016/j.aam.2012.08.009