Kuhlmann (2010) investigates several such classes, based on well-nestedness and gap degree constraints (Bodirsky, Kuhlmann, and Möhl 2005), relating them to lexicalized constituency grammar formalisms.
kuhlmann shows that linear context-free rewriting systems (LCFRS) with fan-out k (Vijay-Shanker, Weir, and Joshi 1987;Satta 1992) induce the set of dependency structures with gap degree at most k − 1; coupled CFG in which the maximal rank of a nonterminal is k (Hotz and Pitsch 1996) induces the set of wellnested dependency structures with gap degree at most k − 1; and finally, LTAG (Joshi and Schabes 1997) induces the set of well-nested dependency structures with gap degree at most 1.