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Finding exact solutions to spin models is a fundamental problem of manybody physics. A workhorse technique for exact solution methods is mapping to an effective description by noninteracting fermions. The paradigmatic example of this is the JordanWigner transformation for finding an exact solution to the onedimensional XY model. Another important example is the exact freefermion solution to the twodimensional Kitaev honeycomb model. I will describe a framework for recognizing freefermionsolvable spin models utilizing the tools of graph theory. Our first main result relies on a connection to the graphtheoretic problem of recognizing line graphs, which has been solved optimally. This characterization reveals a complete set of frustration structures which obstruct a freefermion solution. We further give a classification of the Pauli symmetries that can be present in spin models with such a freefermion solution. I will next give a generalization of this result beyond the setting of the JordanWigner transformation to a family of models whose associated graphs contain neither claws nor even holes. We expect this characterization to motivate a renewed exploration of freefermionsolvable models. Host: Gopi Muraleedharan 