The complete set of minimal simple graphs that support unsatisfiable 2-CNFs
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A propositional logic sentence in conjunctive normal form that has clauses with two literals (a 2-CNF) can be associated with a multigraph in which the vertices correspond to the variables and edges to clauses. We first show that every such sentence that has been reduced, that is, which is unchanged under 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 can be contracted to a graph that contains one of three specific small graphs as a subgraph. The contraction refers to edge-contractions of edges not contained in a triangle. Equivalently, all reduced 2-CNF sentences supported on a given simple graph are satisfiable if and only if those three graphs are forbidden as subgraphs in contractions of the original graph. We conclude with a discussion of why the Robertson-Seymour graph minor theorem does not apply in our approach.
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