Leveraging protein-protein interaction networks to identify groups of proteins and their common functionality is an important problem in bioinformatics. Systems-level analysis of protein-protein interactions is made possible through network science and modeling of high-throughput data. From these analyses, small protein complexes are traditionally represented graphically as complete graphs or dense clusters of nodes. However, there are certain graph theoretic properties that have not been extensively studied in PPI networks, especially as they pertain to cluster discovery, such as planarity. Planarity of graphs have been used to reflect the physical constraints of real-world systems outside of bioinformatics, in areas such as mapping and imaging. Here, we investigate the planarity property in network models of protein complexes. We hypothesize that complexes represented as PPI subgraphs will tend to be planar, reflecting the actual physical interface and limits of components in the complex. When testing the planarity of known complex subgraphs in S. cerevisiae and selected mammalian PPIs, we find that a majority of validated complexes possess this planar property. We discuss the biological motivation of planar versus nonplanar subgraphs, observing that planar subgraphs tend to have longer protein components. Functional classification of planar versus nonplanar complex subgraphs reveals differences in annotation of these groups relating to cellular component organization, structural molecule activity, catalytic activity, and nucleic acid binding. These results provide a new quantitative and biologically motivated measure of real protein complexes in the network model, important for the development of future complex-finding algorithms in PPIs. Accounting for this property paves the way to new means for discovering new protein complexes and uncovering the functionality of unknown or novel proteins.