Idafa in traditional Arabic grammar is an umbrella construction that covers several phenomena including what is expressed in English as noun-noun compounds and Saxon & Norman genitives. Additionally, Idafa participates in some other constructions, such as quantifiers, quasi-prepositions, and adjectives. Identifying the various types of the Idafa construction (IC) is of importance to Natural Language Processing (NLP) applications. Noun-Noun compounds exhibit special behaviour in most languages impacting their semantic interpretation. Hence distinguishing them could have an impact on downstream NLP applications. The most comprehensive computational syntactic representation of the Arabic language is found in the LDC Arabic Treebank (ATB). Despite its coverage, ICs are not explicitly labeled in the ATB and furthermore, there is no clear distinction between ICs of noun-noun relations and other traditional ICs. Hence, we devise a detailed syntactic and semantic typification process of the IC phenomenon in Arabic. We target the ATB as a platform for this classification. We render the ATB annotated with explicit IC labels in addition to further semantic characterization which is useful for syntactic, semantic and cross language processing. Our typification of IC comprises 3 main syntactic IC types: False Idafas (FIC), Grammatical Idafas (GIC), and True Idafas (TIC), which are further divided into 10 syntactic subclasses. The TIC group is further classified into semantic relations. We devise a method for automatic IC labeling and compare its yield against the CATiB Treebank. Our evaluation shows that we achieve the same level of accuracy, but with the additional fine-grained classification into the various syntactic and semantic types.