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Graph 3-Colorability Instance: a graph ๐บ = (๐, ๐ธ). Question: Is there a color assignment ๐: ๐ → {๐, ๐, ๐} such that ๐(๐ฃ๐ ) ≠ ๐(๐ฃ๐ ) if ๐ฃ๐ and ๐ฃ๐ are adjacent? Example 4: 3-SAT → Graph 3-Colorability Given an arbitrary instance of 3-Sat, (i.e. a set of variables ๐ = {๐ข1 , ๐ข2 , โฏ , ๐ข๐ } and a set of size three clauses, = {๐1 , ๐2 , โฏ , ๐๐ } ) we must construct an equivalent instance of 3-Colorability. The construction will be an example of “component design”. Each component will be a subgraph that effectively performs the particular role. Some of the vertices of the graph will correspond to variables. The way those vertices are colored will determine a true/false assignment for the corresponding variables. Example 4: 3-SAT → Graph 3-Colorability Given an arbitrary instance of 3-Sat, (i.e. a set of variables ๐ = {๐ข1 , ๐ข2 , โฏ , ๐ข๐ } and a set of size three clauses, = {๐1 , ๐2 , โฏ , ๐๐ } ) we must construct an equivalent instance of 3-Colorability. 1. Color-to-True/False Correspondence The “center” of our graph will be the triangle below. These vertices do not correspond to variables but one have the color corresponding to “true”, one will have the color corresponding to “false” and the third will be the “neutral” or “neither” or “not like the others” color. We can assume these vertices are colored as below. F T N Example 4: 3-SAT → Graph 3-Colorability Given an arbitrary instance of 3-Sat, (i.e. a set of variables ๐ = {๐ข1 , ๐ข2 , โฏ , ๐ข๐ } and a set of size three clauses, = {๐1 , ๐2 , โฏ , ๐๐ } ) we must construct an equivalent instance of 3-Colorability. ๐ขฬ ๐ F โฐ ๐ขฬ 2 T N ๐ข๐ ๐ข2 ๐ขฬ 1 ๐ข1 2. True/False Determination Now we create ๐ pairs of new vertices. One vertex of the ๐ ๐กโ pair will correspond to variable ๐ข๐ and the other vertex will correspond to ๐ขฬ ๐ . Connect each pair to the “N” vertex above to form a subgraph that ensures that each vertex corresponding to a variable gets colored either blue = “T” or “red = F” and that the vertex corresponding to its negation is colored the opposite. Side Note: Suppose that the edges leading away have forced the green and red colors as shown. Can the rest of the graph on left be 3-colored? How about the graph on the right? Example 4: 3-SAT → Graph 3-Colorability Given an arbitrary instance of 3-Sat, (i.e. a set of variables ๐ = {๐ข1 , ๐ข2 , โฏ , ๐ข๐ } and a set of size three clauses, = {๐1 , ๐2 , โฏ , ๐๐ } ) we must construct an equivalent instance of 3-Colorability. 3. Satisfaction Testing We will create one of the gadgets to the right for each clause, ๐๐ . The vertices shown only appear within one gadget. Arrows indicate connections to vertices already introduced in above components. We must find what the connections need to be so that the subgraph is 3-colorable if and only if the clause is satisfied.