  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\"obner basis can be computed
by studying paths in the graph. Since these Gr\"obner 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.
