We find a condition on the action of a finitely presented group on a simplicial tree which guarantees that this action will be dominated by an acylindrical action with finitely generated edge stabilisers, and find the first example of an action of a finitely presented group where there is no such dominating action. As a consequence, we show that any hyperbolic group that admits a decomposition as an acylindrical graph of (possibly infinitely generated) free groups is virtually cocompact special, and that if one assumes that every finite-height subgroup of a hyperbolic group is quasi-convex then every finitely generated subgroup of a one-relator group with an acylindrical Magnus hierarchy admits a quasi-convex hierarchy.
William Cohen (Wed,) studied this question.