Edge ideals of oriented graphs

Let $mathcal{D}$ be a weighted oriented graph and let $I(mathcal{D})$ be its edge ideal. Under a natural condition that the underlying (undirected) graph of $mathcal{D}$ contains a perfect matching consisting of leaves, we provide several equivalent conditions for the Cohen-Macaulayness of $I(mathcal{D})$. We also completely characterize the Cohen-Macaulayness of $I(mathcal{D})$ when the underlying graph of $mathcal{D}$ is a bipartite graph. When $I(mathcal{D})$ fails to be Cohen-Macaulay, we give an instance where $I(mathcal{D})$ is shown to be sequentially Cohen-Macaulay.