The complete set of minimal simple graphs that support unsatisfiable 2-CNFs

A propositional logic sentence in conjunctive normal form that has clauses of length at most two (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We show that every 2-CNF that has been reduced under the application of certain tautologies, is equisatisfiable to a 2-CNF whose associated multigraph is, in fact, a simple graph. Our main result is a complete characterization of graphs that can support unsatisfiable 2-CNF sentences. We show that a simple graph can support an unsatisfiable reduced 2-CNF sentence if and only if it contains any one of four specific small graphs as a topological minor. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if all subdivisions of those four graphs are forbidden as subgraphs of the original graph.