Properties of symbolic powers of edge ideals of weighted oriented graphs

Let D be a weighted oriented graph and I(D) be its edge ideal. We provide one method to find all the minimal generators of I-subset of C, where C is a maximal strong vertex cover of D and I-subset of C is the intersections of irreducible ideals associated to the strong vertex covers contained in C. If D' is an induced digraph of D, under a certain condition on the strong vertex covers of D' and D, we show that I(D')((s)) not equal I(D')(s) for some s >= 2 implies I(D)((s)) not equal I(D)(s). We provide the necessary and sufficient condition for the equality of ordinary and symbolic powers of edge ideal of the union of two naturally oriented paths with a common sink vertex. We characterize all the maximal strong vertex covers of D such that at most one edge is oriented into each of its vertices and w(x) >= 2 if deg(D)(x) >= 2 for all x is an element of V (D). Finally, if D is a weighted rooted tree with the degree of root is 1 and w(x) >= 2 when deg(D)(x) = 2 for all x is an element of V (D), we show that I(D)((s)) = I(D)(s) for all s >= 2.