Let $\mathcal{K}$ be an abstract class of groups. Suppose $\mathcal{K}$ contains at least a non trivial group.
Then $\mathcal{K}$ is called a root-class if the following conditions are satisfied:
1. If $A \in \mathcal{K}$ and $B \leq A$, then $B \in \mathcal{K}$.
2. If $A \in \mathcal{K}$ and $B \in \mathcal{K}$, then $A\times B \in \mathcal{K}$.
3. If $1\leq C \leq B \leq A$ is a subnormal sequence and $A/B, B/C \in \mathcal{K}$, then there exists a normal subgroup $D$ in group $A$
such that $D \leq C$ and $A/D \in \mathcal{K}$.
Group $G$ is root-class residual (or $\mathcal{K}$-residual), for a root-class $\mathcal{K}$ if,
for every $1 \not = g \in G$,
exists a homomorphism $\varphi $ of group $G$ onto a group of root-class $\mathcal{K}$ such that $g\varphi \not = 1$.
Equivalently, group $G$ is $\mathcal{K}$-residual if, for every $1 \not = g \in G$,
there exists a normal subgroup $N$ of $G$ such that $G/N \in \mathcal{K}$ and $g \not \in N$.
The most investigated residual properties of groups are finite groups residuality (residual finiteness),
$p$-finite groups residuality and soluble groups residuality.
All there three classes of groups are root-classes.
Therefore results about root-class residuality have safficiently enough general character.
Let $\mathcal{K}$ be a root-class of finite groups.
And let $G$ be a fundamental group of a finite graph of groups with finite edges groups.
The necessary and sufficient condition of virtual $\mathcal{K}$-residuality
for the group $G$ is obtained.
Bibliography: 16 titles.
