Some thoughts and comments on the VC to Hamiltonian Circuit (Cycle) transformation.
1) for each edge uv, there is a 12 vertex component created. One "side" consists of "u"
vertices, and the other "v" vertices. A cycle has been formed with all the "u" vertices
from all the edge components constructed with edges having u as one endpoint. A "u
segment" of a proposed Ham. Cycle is a path from one "z" vertex to the first u vertex in
one of these components and proceeds through all the u vertices (all components
containing u vertices), until an exit is made to another "z" vertex.
2) if you enter a component at a "u" vertex, you must leave it from a u vertex (otherwise
you leave some vertices which can not be included in any proposed cycle).
3) when going through a uv component to pick up the u vertices, you may choose to also
pick up the v vertices in the component. Or not, depending upon whether the "v" segment
is to be taken. If the v segment is to be taken (or already has been taken), you can not
include the v's in the u segment – the v cycle would be "broken."
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 Fall '10
 Dutton
 Vertex, Glossary of graph theory, vertices, Graph theory objects, Hamiltonian path, William Rowan Hamilton

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