Implication graph

Implication graph

In mathematical logic, an implication graph is a skew-symmetric directed graph "G"("V", "E") composed of vertex set "V" and directed edge set "E". Each vertex in "V" represents the truth status of a Boolean literal, and each directed edge "e"("u", "v") connecting vertex "u" and vertex "v" represents the implication "If the literal "u" is true then the literal "v" is also true". IGs were originally used for analyzing complex Boolean expressions.

A 2-satisfiability instance in conjunctive normal form can be transformed into an implication graph by replacing each of its disjunctions by a pair of implications. The instance is satisfiable if and only if no literal and its negation belong to the same strongly connected component of the implication graph; this characterization can be used to solve 2-satisfiability instances in linear time. [cite journal|author = Aspvall, Bengt; Plass, Michael F.; Tarjan, Robert E.|title = A linear-time algorithm for testing the truth of certain quantified boolean formulas|journal = Information Processing Letters | volume = 8 | issue = 3 | pages = 121–123|year = 1979|doi = 10.1016/0020-0190(79)90002-4]

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