Graph exhaustion ($V_1\subseteq V_2\subseteq \cdots ,{\bigcup }_n{V}_n=V$)

An exhaustion of a countable graph is an increasing sequence of finite connected graph vertex sets whose union is the whole graph vertex set. In a locally finite graph, every prescribed finite graph path together with all graph neighbours of its visited graph vertices lies in a sufficiently large exhaustion set. This permits coupling finite-volume and infinite-volume simple random walks up to a finite stopping time.