Componentwise linear powers and the $x$-condition

Let $S=K[x_1,\ldots,x_n]$ be the polynomial ring over a field and $A$ standard graded $S$-algebra. In terms of Gröbner basis defining ideal $J$ we give condition, called $x$-condition, which implies that all components $A_k$ have linear quotients with additional assumptions are componentwise linear. A typical example such an algebra is Rees $\mathcal{R}(I)$ or symmetric $\textrm{Sym}(M)$ module $M$. We apply our criterion to study certain algebras powers vertex cover ideals classes graphs.