Planar Formulae and Their Uses

We define the set of planar boolean formulae, and then show that the set of true quantified planar formulae is polynomial space complete and that the set of satisfiable planar formulae is NP-complete. Using these results, we are able to provide simple and nearly uniform proofs of NP-completeness for planar node cover, planar Hamiltonian circuit and line, geometric connected dominating set, and of polynomial space completeness for planar generalized geography. The NP-completeness of planar node cover and planar Hamiltonian circuit and line were first proved elsewhere M. R. Garey and D. S. Johnson, The rectilinear Steiner tree is NP-complete, SIAM J. Appl. Math., 32 (1977), pp. 826–834 and M. R. Garey, D. S. Johnson and R. E. Tarjan, The planar Hamilton circuit problem is NP-complete, SIAM J. Comp., 5 (1976), pp. 704–714.