From any directed graph [Formula: see text] one can construct the graph inverse semigroup [Formula: see text], whose elements, roughly speaking, correspond to paths in [Formula: see text]. Wang and Luo showed that the congruence lattice [Formula: see text] of [Formula: see text] is upper-semimodular for every graph [Formula: see text], but can fail to be lower-semimodular for some [Formula: see text]. We provide a simple characterization of the graphs [Formula: see text] for which [Formula: see text] is lower-semimodular. We also describe those [Formula: see text] such that [Formula: see text] is atomistic, and characterize the minimal generating sets for [Formula: see text] when [Formula: see text] is finite and simple.