Abstract: While mapping class groups of surfaces of finite type are finitely generated, this is not the case with infinite type surfaces. Asymptotic rigidity is a method to obtain finitely generated dense subgroups of such mapping class groups. In joint work with Thomas Hill, Sanghoon Kwak, and Brian Udall, we extend the notion of asymptotic rigidity to the "mapping class groups" of locally finite, infinite graphs (in the sense of Algom-Kfir–Bestvina). When the underlying graph has finitely many ends, this is analogous to a Houghton group. We demonstrate that while many of the standard results for Houghton-type groups hold for our graph Houghton groups, they are not commensurable to any of the extant variants. Additionally, we present an explicit finite presentation in the case where there are at least 3 ends.