Complex dynamical systems often contain many unknown parameters and variables that may or may not serve as contributors to interesting behaviors a system may exhibit. For chemical and biochemical systems, which are typically quite large, these include (but are not limited to) behaviors stemming from bifurcations such as oscillations, patterns and multistationarity (multiple steady states). Due to the size and complexity of these systems, a dynamics-preserving reduction scheme that is able to isolate the necessary contributors to these systems to not only reduce their complexity, but to also reduce the level of uncertainty a system may have---such as unknown parameters and variables---is desired. The purpose of this thesis is to develop alternative reduction methods for (bio)chemical systems that are modeled by mass-action kinetics that are unlike other common techniques that exploit timescales in stiff models or that are optimization-based. I instead look to a graph-based approach by representing the model as a bipartite graph and investigating its subnetworks known as fragments, which correspond directly to terms in the characteristic polynomial. In this representation, I preserve key elements of these bipartite graphs---critical fragments---in order to maintain necessary conditions for behaviors such as positive-feedback oscillations and multistationarity. These results are then applied to an existing model for the transcriptional control of Hmp, an NO detoxifying enzyme, by the iron-sulfur protein FNR that displays bistability. The initial model consists of 15 mass-action reactions and 11 species, and the reduced model ends up with 10 reactions and 7 species.