  We show that every infinite graph which is locally finite and connected
admits a translation-like action by $\mathbb{Z}$ such that the distance between
a vertex $v$ and $v\ast 1$ is uniformly bounded by 3. This action can be taken
to be transitive if and only if the graph has one or two ends. This strengthens
a theorem by Brandon Seward.
  Our proof is constructive, and thus it can be made computable. More
precisely, we show that a finitely generated group with decidable word problem
admits a translation-like action by $\mathbb{Z}$ which is computable, and
satisfies an extra condition which we call decidable orbit membership problem.
  As an application we show that on any finitely generated infinite group with
decidable word problem, effective subshifts attain all effectively closed
Medvedev degrees. This extends a classification proved by Joseph Miller for
$\mathbb{Z}^{d},$ $d\geq1$.

Dynamical Systems;Combinatorics;Group Theory;Logic