Planar 3-SAT with a Clause/Variable Cycle

TL;DR

This paper proves that Planar 3-SAT remains NP-complete even with a specific Hamiltonian cycle restriction on the incidence graph, and it also shows that certain monotone instances are always satisfiable, clarifying previous open questions.

Contribution

It introduces a new graph restriction involving a Hamiltonian cycle in the incidence graph and proves NP-completeness under this restriction, while also resolving the complexity of monotone instances with three variables per clause.

Findings

NP-completeness persists under the new cycle restriction

Monotone instances with three variables per clause are always satisfiable

Clarifies the complexity status of specific Planar 3-SAT variants

Abstract

In the Planar 3-SAT problem, we are given a 3-SAT formula together with its incidence graph, which is planar, and are asked whether this formula is satisfiable. Since Lichtenstein's proof that this problem is NP-complete, it has been used as a starting point for a large number of reductions. In the course of this research, different restrictions on the incidence graph of the formula have been devised, for which the problem also remains hard. In this paper, we investigate the restriction in which we require that the incidence graph can be augmented by the edges of a Hamiltonian cycle that first passes through all variables and then through all clauses, in a way that the resulting graph is still planar. We show that the problem of deciding satisfiability of a 3-SAT formula remains NP-complete even if the incidence graph is restricted in that way and the Hamiltonian cycle is given. This complements previous results demanding cycles only through either the variables or clauses. The problem remains hard for monotone formulas, as well as for instances with exactly three distinct variables per clause. In the course of this investigation, we show that monotone instances of Planar 3-SAT with exactly three distinct variables per clause are always satisfiable, thus settling the question by Darmann, D\"ocker, and Dorn on the complexity of this problem variant in a surprising way.