Componentwise linearity of powers of edge ideals of weighted oriented graphs

In this paper, we study the componentwise linearity of powers of edge ideal of a weighted oriented graph $D$. We give a characterization for componentwise linearity of the edge ideal $I(D)$ in terms of forbidden subgraphs of $D$. If $D$ is house-free or complete $r$-partite, then the following statements are equivalent: (1) $I(D)$ is componentwise linear; (2) $I(D)$ is vertex splittable; (3) $I(D)$ has linear quotient property; (4) both $G$ and $H(I(D)_{(2)})$ are co-chordal and $D_1,D_2,D_3,D_4$ as in Figure 3, are not induced subgraphs of $D$. Furthermore, if $D$ is a complete $r$-partite weighted oriented graph, then we show that: $I(D)^k$ is componentwise linear, for some $k\geq 2 \iff I(D)$ is componentwise linear.