Toric ideals of weighted oriented graphs

Given a vertex-weighted oriented graph, we can associate to it set of monomials. We consider the toric ideal whose defining map is given by these find generating for certain classes graphs which depends on combinatorial structure and weights graph. provide result analogous unweighted, unoriented graph case, show that when associated simple has only trivial even closed walks, zero ideal. Moreover, give necessary sufficient conditions weighted be generated single binomial describe in terms