Biological networks are fast becoming a popular tool for modeling high-throughput data, especially due to the ability of the network model to readily identify structures with biological function. However, many networks are fraught with noise or coincidental edges, resulting in signal corruption. Previous work has found that the implementation of network filters can reduce network noise and size while revealing significant network structures, even enhancing the ability to identify these structures by exaggerating their inherent qualities. In this study, we implement a hybrid network filter that combines features from a spanning tree and near-chordal subgraph identification to show how a filter that incorporates multiple graph theoretic concepts can improve upon network filtering. We use three different clustering methods to highlight the ability of the filter to maintain network clusters, and find evidence that suggests the clusters maintained are of high importance in the original unfiltered network due to high-degree and biological relevance (essentiality). Our filter highlights the advantages of integration of graph theoretic concepts into biological network analysis.