2-Sat Sub-Clauses and the Hypernodal Structure of the 3-Sat Problem

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Like simpler graphs, nested (hypernodal) graphs consist of two components: a set of nodes and a set of edges, where each edge connects a pair of nodes. In the hypernodal graph model, however, a node may contain other graphs, so that a node may be contained in a graph that it contains. The inherently recursive structure of the hypernodal graph model aptly characterizes both the structure and dynamic of the 3-sat problem, a broadly applicable, though intractable, computer science problem. In this paper I first discuss the structure of the 3-sat problem, analyzing the relation of 3-sat to 2-sat, a related, though tractable problem. I then discuss sub-clauses and sub-clause thresholds and the transformation of sub-clauses into implication graphs, demonstrating how combinations of implication graphs are equivalent to hypernodal graphs. I conclude with a brief discussion of the use of hypernodal graphs to model the 3-sat problem, illustrating how hypernodal graphs model both the conditions for satisfiability and the process by which particular 3-sat assignments either succeed or fail.