  We propose a definition of graph subshifts of finite type that can be seen as
extending both the notions of subshifts of finite type from classical symbolic
dynamics and finitely presented groups from combinatorial group theory. These
are sets of graphs that are defined by forbidding finitely many local patterns.
In this paper, we focus on the question whether such local conditions can
enforce a specific support graph, and thus relate the model to classical
symbolic dynamics. We prove that the subshifts that contain only infinite
graphs are either aperiodic, or feature no residual finiteness of their period
group, yielding non-trivial examples as well as two natural undecidability
theorems.
