Boundary properties of the satisfiability problems

The satisfiability problem is known to be NP-complete in general and for many restricted instances, such as CNF formulas with at most 3 variables per clause and at most 3 occurrences per variable, or planar formulas. The latter example refers to graphs representing satisfiability instances. These are bipartite graphs with vertices representing clauses and variables, and edges connecting variables to the clauses containing them. Finding the strongest possible restrictions under which the problem remains NP-complete is important for at least two reasons. First, this can make it easier to establish the NP-completeness of new problems by allowing easier transformations. Second, this can help clarify the boundary between tractable and intractable instances of the problem. In this paper, we address the second issue and reveal the first boundary property of graphs representing satisfiability instances