Planarity conditions for Planar 1-in-3 SAT Planar 3SAT is NP-complete. A planar 3SAT instance is a 3SAT instance for which the graph built using the following rules is planar:


add a vertex for every $x_i$ and $\bar{x_i}$
add a vertex for every clause $C_j$
add an edge for every $(x_i,\bar{x_i})$ pair
add an edge from vertex $x_i$ (or $\bar{x_i}$) to each vertex that represent a clause that contains it
add edges between two consecutive variables  $(x_1,x_2),(x_2,x_3),...,(x_n,x_1)$


In particular, rule 5 builds a "backbone" that splits the clauses in two distinct regions.

Planar 1-in-3 SAT is NP-complete, too.


But for planar 1-in-3 SAT are the planarity conditions defined in the same way as in Planar 3SAT ? In particular, can we assume that there is a backbone that links the variables $(x_i,x_{i+1})$  ?
Yes you can. Actually you can even show that something stronger is true. The problem know as Positive Planar 1-in-3-SAT is NP-complete as shown by Mulzer and Rote. 

In this version of 1-in-3-SAT, you require for every input formula that


you have three variables per clause, none of them negated
the graph of the formula is planar, even if you add the "backbone" between the variable vertices


The reduction is from Planar 3-SAT.