  Any graph can be represented pictorially as a figure. Moreover, it can be
represented as two or more figures that can be have different properties to
each other. For the purpose of HCP, we represent a graph by two such figures.
In each of them, there is an exterior part called the contour, and an interior
part. These two figures differ from each other by the constitution of the edges
in the interior part. That is, any edges in the interior part for one figure
are not in the interior for the other figure. We call these two figures basic
objects. We develop rules and algorithms that allow us to represent any graph
of degree d <= 3 by two basic objects. Individually, neither of these
representations possess the features to easily determine the Hamiltonicity of
the graph. However, the combination of these two figures, once certain weights
are assigned to their edges, allows us to determine the Hamiltonicity with a
polynomial-time check. The rules for the assignment of weights are: 1. The
weight of any edge of the interior part is 0, for both objects. 2. In both
figures any common edge of the contour has the same weight. The weights of the
edges allow us to extend the number of parameters of the objects, that is
sufficient to determine the Hamiltonicity of the graph. Then, if the graph is
Hamiltonian, then both figures possess the same set of parameters. If the sets
of parameters for two figures are different, then the graph is not Hamiltonian.
The parameters that determine the Hamiltonicity of the graph are the sums of
weights of edges and windows of contours in the figure. The algorithms of their
construction do not contain a combinatorial number of elements and have
polynomial complexity. We also supply an estimate of the complexity of each
algorithm.
