{{short description|Directed graph representing a Boolean expression}} [[Image:Implication graph.svg|thumb|upright=1.35|An implication graph representing the {{nowrap|2-satisfiability}} instance <math>\scriptscriptstyle (x_0\lor x_2)\land(x_0\lor\lnot x_3)\land(x_1\lor\lnot x_3)\land(x_1\lor\lnot x_4)\land(x_2\lor\lnot x_4)\land{}\atop\quad\scriptscriptstyle(x_0\lor\lnot x_5)\land (x_1\lor\lnot x_5)\land (x_2\lor\lnot x_5)\land (x_3\lor x_6)\land (x_4\lor x_6)\land (x_5\lor x_6).</math>]]
In mathematical logic and graph theory, an '''implication graph''' is a skew-symmetric, directed graph {{math|1=''G'' = (''V'', ''E'')}} composed of vertex set {{mvar|V}} and directed edge set {{mvar|E}}. Each vertex in {{mvar|V}} represents the truth status of a Boolean literal, and each directed edge from vertex {{mvar|u}} to vertex {{mvar|v}} represents the material implication "If the literal {{mvar|u}} is true then the literal {{mvar|v}} is also true". Implication graphs were originally used for analyzing complex Boolean expressions.
==Applications== 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. For example, the statement <math>(x_0\lor x_1)</math> can be rewritten as <math>(\neg x_0 \rightarrow x_1)</math>, but <math> (\neg x_1 \rightarrow x_0)</math> also works. An instance is satisfiable if and only if no literal and its negation belong to the same strongly connected component of its implication graph; this characterization can be used to solve {{nowrap|2-satisfiability}} instances in linear time.<ref>{{cite journal|author1= Aspvall, Bengt |author2=Plass, Michael F. |author3=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}}</ref>
In CDCL SAT-solvers, unit propagation can be naturally associated with an implication graph that captures all possible ways of deriving all implied literals from decision literals,<ref>{{cite conference|author1=Paul Beame |author2=Henry Kautz |author3=Ashish Sabharwal |title = Understanding the Power of Clause Learning| conference = IJCAI | pages = 1194–1201|year = 2003|url=https://www.cs.cornell.edu/~sabhar/publications/learnIJCAI03.pdf}}</ref> which is then used for clause learning.
==References== <references/>
Category:Boolean algebra Category:Application-specific graphs Category:Directed graphs Category:Graph families