On conditions for the approximability of the fundamental groups of graphs of groups by root classes of groups

Suppose that $\mathrm{\Gamma }$ is a non-empty connected graph, $\mathfrak{G}$ is the fundamental group of a graph of groups over $\mathrm{\Gamma }$, and $\mathcal{C}$ is a root class of groups (the last means that $\mathcal{C}$ contains non-trivial groups and is closed under taking subgroups, extensions, and Cartesian powers of a certain type). It is known that $\mathfrak{G}$ is residually a $\mathcal{C}$-group if it has a homomorphism onto a group from $\mathcal{C}$ acting injectively on all vertex groups. We prove that, in this assertion, the words “vertex groups” can be replaced by “edge subgroups” provided all vertex groups are residually $\mathcal{C}$-groups. We also show that the converse doesn’t need to hold if $\mathcal{C}$ consists of periodic groups and contains at least one infinite group.