The classification/categorization of oscillatory chemical reactions and the determination of the connectivity of species in a reaction mechanism can be deduced from several experiments. In this article we show the same for bifurcation diagrams. We construct such diagrams for skeletal models of each of the known categories; these are distinct and can be used for the classification of species essential for oscillations and for the categorization of oscillatory reactions. The bifurcation diagrams are closely related to the concentration shift matrix. Prototypes of categories 1B and 1C are extended by adding nonessential species. By assuming that in a flow-through stirred reactor bifurcation diagrams for each pair of constraints (the inflow concentrations and the flow rate) can be measured, we are able to determine the category of the examined oscillator and also deduce the connectivity of the corresponding reaction network. Bifurcation diagrams possess a cusp region with specific tilt. This information together with the knowledge of a concentration shift across the saddle-node bifurcation provide the clues for categorization as well as for the reconstruction of the network's connectivity. As an example, we present an analysis of the Belousov-Zhabotinsky reaction represented by the Field-Koros-Noyes mechanism and discuss the feasibility of reconstruction of the mechanism from experiments.