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An exhaustion of a countable graph is an increasing sequence $V_0\subseteq V_1\subseteq \cdots$ of finite vertex sets whose union is the whole vertex set. Enumerating the vertices and taking finite initial segments gives one. The induced finite subgraphs need not be connected graph.

For a connected graph that is a locally finite graph, the graph balls about a fixed vertex give a connected exhaustion: each ball is finite, and every vertex has finite graph distance from the centre. In a locally finite graph, any prescribed finite graph path, together with all graph neighbours of its visited 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. A disconnected graph cannot have an exhaustion by connected vertex sets covering different connected components of a graph.

Let $G$ be a group, $H\le G$ a subgroup and $S$ a generating set. The Schreier graph has left cosets $G/H$ as vertices and a directed edge $gH\to sgH$ labeled $s$ for each $s\in S$. Left multiplication makes this independent of the chosen representative of $gH$.

When $S$ is symmetric, the radius-$m$ graph ball about $H$ consists ofThese are cosets reached by words of length at most $m$. The single exact power $S^m$ gives the same set when $1\in S$; without this padding convention it may miss vertices reached at shorter lengths. For nonsymmetric $S$, the directed reachability version uses the displayed words, while the underlying undirected graph uses $S\cup S^{-1}$.

For $H=\{1\}$, this recovers the left-multiplication Cayley graph. More generally the Schreier graph records the group action on the set of cosets, including the stabilizer $H$.