Relating planar graph drawings to planar satisfiability problems

A SAT graph $G\left(\Phi \right)$ of a satisfiability instance Φ consists of a vertex for each clause and a vertex for each variable, where there exists an edge between a clause vertex and a variable vertex if and only if the variable or its negation appears in that clause. Many satisfiability problems, which are NP-hard, become polynomial-time solvable when the SAT graph is restricted to satisfy some graph properties. A rich body of research attempts to narrow down the boundary between the NP-hardness and polynomial-time solvability of various satisfiability problems. In this paper, we examine planar satisfiability problems and leverage planar graph drawing algorithms to improve our understanding of these problems. A rich body of graph drawing algorithms exists to check whether a planar graph admits a drawing that satisfies certain drawing aesthetics. We show how the existing graph drawing knowledge could be used to establish sufficient conditions for a SAT instance to always be satisfiable and give algorithms to efficiently find a satisfying truth assignment. In some cases, our algorithm can find a truth assignment by setting a small number of variables to true, which relates to the satisfiability variants that seek to minimize the number of ones.