  For positive integers $d<n$, let $[n]_d=\{A\in 2^{[n]}\mid |A|=d\}$ where
$[n]=:\{1,2,\ldots, n\}$. For a pure $f$-simplicial complex $\Delta$ such that
${\rm dim}(\Delta)={\rm dim}(\Delta^c)$ and $\mathcal{F}(\Delta)\cap
\mathcal{F}(\Delta^c)=\emptyset$, we prove that the facet ideal $I(\Delta)$ is
Cohen-Macaulay if and only if it has linear resolution. For a $d$-dimensional
pure $f$-simplicial complex $\Delta$ such that $\Delta'=:\langle F\mid F\in
[n]_d\smallsetminus \mathcal F(\Delta)\rangle$ is an $f$-simplicial complex, we
prove that $I(\Delta^c)$ is Cohen-Macaulay if and only if $I(\Delta')$ has
linear resolution.
