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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Vertex, Glossary of graph theory, vertices, Graph theory objects, Hamiltonian path, William Rowan Hamilton