We are interested in a graph-based Knowledge Representation (KR) formalism that would allow for the representation, manipulation, query, and reasoning over dependency structures, and linguistic knowledge of the lexicon in the Meaning-Text Theory framework. Neither the semantic web formalisms nor the conceptual graphs appear to be suitable for this task, and this led to the introduction of the new Unit Graphs (UG) framework. In this paper we will overview the foundational concepts of this framework: the UGs are defined over a UG-support that contains: i) a hierarchy of unit types which is strongly driven by the actantial structure of unit types, ii) a hierarchy of circumstantial symbols, and iii) a set of unit identifiers. Based on these foundational concepts and on the definition of UGs, this paper justifies the use of a deep semantic representation level to represent meanings of lexical units. Rules over UGs are then introduced, and lexicographic definitions of lexical units are added to the hierarchy of unit types. Finally this paper provides UGs with semantics (in the logical sense), and pose the entailment problem, so as to enable the reasoning in the UGs framework.