  For a simple graph $G$, assume that $J(G)$ is the vertex cover ideal of $G$
and $J(G)^{(s)}$ is the $s$-th symbolic power of $J(G)$. We prove that
$(J(C)^{(s)})=(J(C)^s)$ for all $s\geq 1$ and for all odd cycle $C$. For a
simplicial complex $\Delta$, we show that $\Delta$ is vertex decomposable if
$I_{\Delta}^{\vee}$ is weakly polymatroidal. Let $W=G^{\pi}$ be a fully
clique-whiskering graph, we prove that $J(W)^s$ is weakly polymatroidal for all
$s\geq 1$.
