Generalized direct products of groups and their application to the study of residuality of free constructions of groups

We introduce the construction of a generalized direct product associated with a graph of groups and prove two sufficient conditions for its existence. These results are applied to obtain some sufficient conditions for an $\mathrm{HNN}$-extension with central associated subgroups to be residually a $\mathcal{C}$-group where $\mathcal{C}$ is a root class of groups. In particular, it is proved that an $\mathrm{HNN}$-extension of a solvable group with central associated subgroups is residually solvable.