  In this paper we consider a large family of graphs of hierarchically
hyperbolic groups (HHG) and show that their fundamental groups admit HHG
structures. To do that, we will investigate the notion of hierarchical quasi
convexity and show that for a hyperbolic HHS it coincides with the notion of
quasi-convexity. The main technical result, for which we expect further
applications, is that it is possible to incorporate the HHG structure of a
hierarchically hyperbolically embedded subgroup into the HHG structure of the
ambient group. This generalizes and provides some additional details to a
procedure described in [DHS17].
