Cohen-Macaulay weighted oriented edge ideals and its alexander dual

The study of the edge ideal I(DG) of a weighted oriented graph DG with underlying graph G started in the context of Reed-Muller type codes. We generalize a Cohen-Macaulay construction for I(DG), which Villarreal gave for edge ideals of simple graphs. We use this construction to classify all the Cohen-Macaulay weighted oriented edge ideals, whose underlying graph is a cycle. We show that the conjecture on Cohen-Macaulayness of I(DG), proposed by Pitones et al. (2019), holds for I(DCn), where Cn denotes the cycle of length n. Miller generalized the concept of Alexander dual ideals of square-free monomial ideals to arbitrary monomial ideals, and in that direction, we study the Alexander dual of I(DG) and its conditions to be Cohen-Macaulay.