The following appears to be a little hand waving. But it's not really. It is,
though,
necessary to sit down and draw some pictures and follow it
through. Not all "proofs" can be simply "read." Some take work to
follow.
3–SAT
Vertex Cover (VC)
Let I
3SAT
be an arbitrary instance of 3SAT. For integers n and m, U =
{u
1
, u
2
, …, u
n
} and C
i
= [z
i1
, z
i2
, z
i3
} for 1 ≤ i ≤ m, where each z
ij
is either
a u
k
or u
k
' for some k.
Construct an instance of VC as follows.
For 1 ≤ i ≤ n construct 2n vertices, u
i
and u
i
' with an edge between
them.
For each clause C
i
= [z
i1
, z
i2
, z
i3
}, 1 ≤ i ≤ m, construct three vertices z
i1
,
z
i2
, and z
i3
and form a "triangle on them. Each z
ij
is one of the Boolean
variables u
k
or it's complement u
k
'. Draw an edge between z
ij
and the
Boolean variable (whichever it is) Each z
ij
has degree 3. Finally, set k =
n+2m.
Theorem.
The
given instance of 3SAT is satisfiable if and only if the
constructed instance of VC has a vertex cover with at most k vertices.
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 Fall '10
 Dutton
 Logic, Grammatical conjunction, triangle, vertex cover, The Edge, vertices

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