  We generalize theorems of Kesten and Deuschel-Pisztora about the
connectedness of the exterior boundary of a connected subset of $\mathbb{Z}^d$,
where "connectedness" and "boundary" are understood with respect to various
graphs on the vertices of $\mathbb{Z}^d$. We provide simple and elementary
proofs of their results. It turns out that the proper way of viewing these
questions is graph theory, instead of topology.
