Bondary-connectivity via graph theory

Abstract

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.

8 pages. Title of earlier version was: Some short proofs for connectedness of boundaries

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