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 Fr\"oberg's theorem for weighted oriented graphs. We give combinatorial characterizations of componentwise linearity of $I(D)$ if $V^+$ are sinks or $\vert V^+ \vert\leq 1$. Furthermore, if $G$ is chordal or bipartite or $V^+$ are sinks or $\vert V^+ \vert\leq 1$, then we show the following equivalence for $I(D)$: $$\text{Vertex splittable}\,\, \Longleftrightarrow\,\, \text{Linear quotient}\,\, \Longleftrightarrow\,\, \text{Componentwise linear}.$$