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2-satisfiability
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===The set of all solutions=== [[File:2SAT median graph.svg|thumb|upright=1.35|The [[median graph]] representing all solutions to the example 2-satisfiability instance whose implication graph is shown above.]] The set of all solutions to a 2-satisfiability instance has the structure of a [[median graph]], in which an edge corresponds to the operation of flipping the values of a set of variables that are all constrained to be equal or unequal to each other. In particular, by following edges in this way one can get from any solution to any other solution. Conversely, any median graph can be represented as the set of solutions to a 2-satisfiability instance in this way. The median of any three solutions is formed by setting each variable to the value it holds in the [[majority function|majority]] of the three solutions. This median always forms another solution to the instance.<ref>{{citation | last1 = Bandelt | first1 = Hans-Jürgen | last2 = Chepoi | first2 = Victor | contribution = Metric graph theory and geometry: a survey | doi = 10.1090/conm/453/08795 | mr = 2405677 | pages = 49–86 | publisher = American Mathematical Society | location = Providence, RI | series = Contemporary Mathematics | title = Surveys on discrete and computational geometry | volume = 453 | year = 2008| isbn = 978-0-8218-4239-3 }}. {{citation | first1 = F. R. K. | last1 = Chung | author-link1 = Fan Chung | first2 = R. L. | last2 = Graham | author-link2 = Ronald Graham | first3 = M. E. | last3 = Saks | title = A dynamic location problem for graphs | url = http://www.math.ucsd.edu/~fan/mypaps/fanpap/101location.pdf | journal = [[Combinatorica]] | volume = 9 | year = 1989 | pages = 111–132 | doi = 10.1007/BF02124674 | issue = 2| s2cid = 5419897 }}. {{citation | last = Feder | first = T. | title = Stable Networks and Product Graphs | series = Memoirs of the American Mathematical Society | volume = 555 | year = 1995}}.</ref> {{harvtxt|Feder|1994}} describes an algorithm for efficiently listing all solutions to a given 2-satisfiability instance, and for solving several related problems.<ref>{{citation|first=Tomás|last=Feder|title=Network flow and 2-satisfiability|journal=[[Algorithmica]]|volume=11|issue=3|year=1994|pages=291–319|doi=10.1007/BF01240738|s2cid=34194118}}.</ref> There also exist algorithms for finding two satisfying assignments that have the maximal [[Hamming distance]] from each other.<ref>{{citation|first1=Ola|last1=Angelsmark|first2=Johan|last2=Thapper|contribution=Algorithms for the maximum Hamming distance problem|title=Recent Advances in Constraints|publisher=Springer-Verlag|series=Lecture Notes in Computer Science|volume=3419|year=2005|pages=[https://archive.org/details/recentadvancesin0000join/page/128 128–141]|doi=10.1007/11402763_10|isbn=978-3-540-25176-7|url=https://archive.org/details/recentadvancesin0000join/page/128}}.</ref>
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