Componentwise linearity of powers of cover ideals

Let G be a finite simple graph and J (G) denote its vertex cover ideal in a polynomial ring over a field. The k-th symbolic power of J (G) is denoted by J (G)((k)). In this paper, we give a criterion for cover ideals of vertex decomposable graphs to have the property that all their symbolic powers are not componentwise linear. Also, we give a necessary and sufficient condition on G so that J (G)(k) is a componentwise linear ideal for some (equivalently, for all) k >= 2 when G is a graph such that G\N-G[A] has a simplicial vertex for any independent set A of G. Using this result, we prove that J (G)(k) is a componentwise linear ideal for several classes of graphs for all k >= 2. In particular, if G is a bipartite graph, then J (G) is a componentwise linear ideal if and only if J (G)(k) is a componentwise linear ideal for some (equivalently, for all) k >= 2.