Description2SAT median graph.svg	A median graph representing the set of solutions to a 2-satisfiability instance
Date	2 May 2008
Source	Own work
Author	David Eppstein
Permission (Reusing this file)	Public domainPublic domainfalsefalse I, the copyright holder of this work, release this work into the public domain. This applies worldwide. In some countries this may not be legally possible; if so: I grant anyone the right to use this work for any purpose, without any conditions, unless such conditions are required by law.

Detailed description

This graph is formed from the 2-satisfiability instance

${\displaystyle \scriptstyle (x_{0}\lor x_{2})\land (x_{0}\lor \lnot x_{3})\land (x_{1}\lor \lnot x_{3})\land (x_{1}\lor \lnot x_{4})\land (x_{2}\lor \lnot x_{4})\land {} \atop \scriptstyle \quad (x_{0}\lor x_{5})\land (x_{1}\lor x_{5})\land (x_{2}\lor x_{5})\land (x_{3}\lor x_{6})\land (x_{4}\lor x_{6})\land (x_{5}\lor x_{6})}$

by creating a vertex for each satisfying truth assignment to the formula and an edge between any two assignments that differ in the value of a single variable. The vertices in the drawing are labeled by the sequence of variable values,

${\displaystyle \scriptstyle x_{0}x_{1}x_{2}x_{3}x_{4}x_{5}x_{6}.}$

In general, any 2-satisfiability instance has a set of solutions with the structure of a median graph, in which the median of any three solutions is formed by taking a majority vote separately for each variable's value. However, for some 2-satisfiability problems, the edges in this median graph may connect pairs of solutions that differ by simultanously flipping several variables that are forced by the instance to all be equal or unequal.