Discrete time positive systems have been the object of widespread interest in the literature. In the last decade particular attention has been devoted to the analysis and characterization of the different notions of reachability and controllability which, in this context, make their appearance. In this paper, the properties of (ordinary and essential) controllability and reachability are analysed. Based on a graph-theoretic approach, we provide practical criteria to verify whether a given positive system (F, G) is endowed with these properties, together with canonical forms for describing reachable/controllable pairs.