  We investigate the disconnection time of a simple random walk in a discrete
cylinder with a large finite connected base. In a recent article of A. Dembo
and the author it was found that for large $N$ the disconnection time of
$G_N\times\mathbb{Z}$ has rough order $|G_N|^2$, when
$G_N=(\mathbb{Z}/N\mathbb{Z})^d$. In agreement with a conjecture by I.
Benjamini, we show here that this behavior has broad generality when the bases
of the discrete cylinders are large connected graphs of uniformly bounded
degree.
