This paper is the continuation of the research of the author and his
colleagues of the {\it canonical} decomposition of graphs. The idea of the
canonical decomposition is to define the binary operation on the set of graphs
and to represent the graph under study as a product of prime elements with
respect to this operation. We consider the graph together with the arbitrary
partition of its vertex set into n subsets (n-partitioned graph). On the
set of n-partitioned graphs distinguished up to isomorphism we consider the
binary algebraic operation βHβ (H-product of graphs), determined by the
digraph H. It is proved, that every operation βHβ defines the unique
factorization as a product of prime factors. We define H-threshold graphs as
graphs, which could be represented as the product βHβ of one-vertex
factors, and the threshold-width of the graph G as the minimum size of H
such, that G is H-threshold. H-threshold graphs generalize the classes of
threshold graphs and difference graphs and extend their properties. We show,
that the threshold-width is defined for all graphs, and give the
characterization of graphs with fixed threshold-width. We study in detail the
graphs with threshold-widths 1 and 2