This paper scrutinizes various dependency based representations of the syntax of function words, such as prepositions. The focus is on the underlying formal object used to encode the linguistic analyses and its relation to the corresponding linguistic theory. The poly-graph structure is introduced: it consists of a generalization of the concept of graph that allows edges to be vertices of other edges. Such a structure is used to encode dependency-based analyses that are founded on two kinds of morphosyn-tactic criteria: presence constraints and distributional constraints.