# Definition of Planar 3-SAT

What is the standard definition of Planar 3-SAT? I have seen a number of different definitions. What was the original paper that defined it and proved it to be NP-complete?

• What did you find confusing about the results? – Niel de Beaudrap Jul 22 '14 at 16:24
• I see different definitions, like some says: The bipartite graph between clauses and the literals must be planar (i don't know by literals do they mean only x_i's or both x_i and its negation, I mean i don't know what is their gadget graph exactly here?). Some other define two types for it: only the bipartite edges between clauses and literals, or these plus (x_i, ~x_i). Or some other says, the above graph plus the (x_i,x_{i+1})'s? I can not even find the original paper published on it? Basically I can not find a good reference with a perfect definition for it? – user24175 Jul 22 '14 at 16:43
• The original reference is: D. Lichtenstein, "Planar formulae and their uses" (1982); but there are many small variations that are still NP-complete (the NPC proof of most of them is easy). – Marzio De Biasi Jul 22 '14 at 18:02
• @Marzio De Biasi Thank you so much! But, based on this paer, planar 3-SAT is the case that the bipartite graph between the clauses that literals (only x_i's not their negations) is planar. Right? We can easily conclude the case that we include also the negation of x_i's just by adding an edge between them, without disturbing the planarity, right? – user24175 Jul 22 '14 at 18:38
• @tinLoaf: as cited in the very good lecture linked by David Eppstein in his answer, you can look at Mark de Berg and Amirali Khosravi, Optimal Binary Space Partitions in the Plane; in which it is proved that monotone planar 3-SAT is NPC: variables are placed on a horizontal line, all positive clauses are drawn above, all negative clauses are drawn below; in that representation it is easy to replace each variable $x_i$ with two stacked (and also linked) literals, the positive literal $+x_i$ above, the negative literal $-x_i$ below, without breaking planarity condition. – Marzio De Biasi Jan 14 '16 at 21:47