Planar SAT

A planar graph is a graph that can be drawn on the plane in a way such that no two of its edges cross each other. Every 3SAT problem can be converted to an incidence graph in the following manner: For every variable ${\displaystyle v_{i}}$, the graph has one corresponding ${\displaystyle v_{i}}$, and for every clause ${\displaystyle c_{j}}$, the graph has one corresponding node ${\displaystyle c_{j}.}$We create an edge ${\displaystyle (v_{i},c_{j})}$ between variable ${\displaystyle v_{i}}$and clause ${\displaystyle c_{j}}$ whenever ${\displaystyle v_{i}}$ or ${\displaystyle {\bar {v_{i}}}}$ is in ${\displaystyle c_{j}}$. We distinguish positive and negative literals using edge colorings.

Planar 3SAT is a subset of 3SAT in which the incidence graph of the variables and the clauses of a Boolean formula is planar. It is important because it is a restricted variant, and is still NP-complete. Many problems (for example games and puzzles) cannot represent non-planar graphs. Hence, Planar 3SAT provides a way to prove those games to be NP-hard.

Trivially, PLANAR 3SAT is in NP. It is thus sufficient to show that it is NP-hard via reduction from 3SAT.

First, draw the incidence graph of the 3SAT formula. Since no two variables or clauses are connected, the resulting graph will be bipartite. The resulting graph may not be planar. Replace every crossing of edges with a crossover gadget shown here. However, the figure leads to a minor error—some clauses contain 4 variables and some contain only 2 variables so the premises of 3SAT are not followed exactly in the gadget shown. This glitch is easily fixable: For a clause that only contains 2 variables, either create parallel edges from one variable to the clause or create a separate false variable to include in the constraint.

For the 4 variable clause, borrow the reduction from 4SAT to 3SAT to create a gadget that involves introducing an extra variable set to false which represents whether the left or right side of the original clause contained the satisfying literal. This completes the reduction.

• Planar rectilinear 3SAT: Each variable is a horizontal segment on the x-axis while each clause is a horizontal segment above the x-axis with vertical connections to the 3 variables it includes on the x-axis. The connections can either be positive or negative. This problem is NP-complete.[1]
• Planar monotone rectilinear 3SAT: This is a variation of planar rectilinear 3SAT where all positive clauses are above the x-axis and all negative clauses are below the x-axis. This problem is also NP-complete.[2]
• Planar 1-in-3SAT: This is the planar equivalent of 1-in-3SAT. It is also NP-complete.[3]
• Planar NAE 3SAT: This problem is planar equivalent of NAE 3SAT. Surprisingly, it can be solved in polynomial time.[4]