Componentwise linearity of edge ideals of weighted oriented graphs

In this paper, we study the componentwise linearity of edge ideals of weighted oriented graphs. We show that if $D$ is a weighted oriented graph whose edge ideal $I(D)$ is componentwise linear, then the underlying simple graph $G$ of $D$ is co-chordal. This is an analogue of Fröberg’s theorem for weighted oriented graphs. We give combinatorial characterizations of componentwise linearity of $I(D)$ if $V^{+}$ are sinks or $\mathrm{\mid }{V}^{+}\mathrm{\mid }\le 1$. Furthermore, if $G$ is chordal or bipartite or $V^{+}$ are sinks or $\mathrm{\mid }{V}^{+}\mathrm{\mid }\le 1$, then we show the following equivalence for $I(D)$: $$\text{Vertex splittable}⟺\text{Linear quotient}⟺\text{Componentwise linear}\mathrm{.}$$