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 Gr��bner basis can be computed by studying paths in the graph. Since these Gr��bner 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.

6 pages. arXiv admin note: substantial text overlap with arXiv:1110.1338