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Suppose that $\Gamma$ is a non-empty connected graph, $\mathfrak{G}$ is thefundamental group of a graph of groups over $\Gamma$, and $\mathcal{C}$ is aroot class of groups (the last means that $\mathcal{C}$ contains non-trivialgroups and is closed under taking subgroups, extensions, and Cartesian powersof 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, thewords "vertex groups" can be replaced by "edge subgroups" provided all vertexgroups are residually $\mathcal{C}$-groups. We also show that the conversedoesn't need to hold if $\mathcal{C}$ consists of periodic groups and containsat least one infinite group.

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