With the increasing availability of various types of biological data and the ability to measure interrelationships among molecular elements, biological networks have quickly emerged as the go-To structure to model biological elements and relationships. However, there is not a large body of research that closely analyzes the properties of the various biological networks in ways that allow for the increased extraction of valuable information from these networks and establishes useful connections between network structures and corresponding biological properties. In particular, exploring the underlying graph properties of biological networks augments our understanding of biological organisms as complex systems. Understanding these properties is critical to the process of generating knowledge from biological network models. These properties become particularly interesting when they can be correlated with specific structural and functional qualities associated with the entities represented by the graph/network. Planarity may be especially important to understanding and identifying protein complexes, which are frequently subject to physical constraints that may prevent the constitutive protein components from interacting in such a way that the resulting graph abstraction is densely connected. In this work, we investigate the planarity of domain-domain interaction (DDI) graphs for S. cerevisiae protein complexes with validated three-dimensional structures. We found that the majority of these protein complexes were planar, even with the exclusion of complexes that had small DDI graphs with very few edges. We also found significant structural and functional differences between groups of complexes with planar and nonplanar DDI graphs. These results provide additional context for the study of protein complexes within the network model, and this additional context may be important for general knowledge generation, as well as for specific tasks like protein complex identification.