Planar 3-SAT with a clause/variable cycle

Alexander Pilz*

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingConference paperpeer-review


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, di erent 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 is 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 and instances with exactly three distinct variables per clause. In the course of this investigation, we show that monotone instances of Planar 3-SAT with three distinct variables per clause are always satisfiable, thus settling the question by Darmann, Döcker, and Dorn on the complexity of this problem variant in a surprising way.

Original languageEnglish
Title of host publication16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018
PublisherSchloss Dagstuhl - Leibniz-Zentrum für Informatik
Number of pages2803
ISBN (Electronic)9783959770682
Publication statusPublished - 1 Jun 2018
Externally publishedYes
Event16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018 - Malmo, Sweden
Duration: 18 Jun 201820 Jun 2018


Conference16th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2018


  • 1-in-3-SAT
  • Phrases 3-SAT
  • Planar graph

ASJC Scopus subject areas

  • Software


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