H-product of graphs, H-threshold graphs and threshold-width of graphs

One method of graph decomposition is to define a binary operation on the set of graphs and to represent graphs as products of prime elements with respect to this operation. We consider a graph together with a partition of its vertex set into n subsets ( n -partitioned graph). On the set of all isomorphism classes of n -partitioned graphs we consider a binary algebraic operation ∘ H ( H -product of graphs), determined by a digraph H . We prove that every operation ∘ H defines the unique factorization as a product of prime factors. We define H -threshold graphs as graphs that can be represented as products of one-vertex factors under the operation ∘ H . Threshold-width of a graph G is the minimum number of vertices in a digraph H such that G is an H -threshold graph. In particular, threshold graphs and difference graphs are H -threshold graphs for certain digraphs H . Threshold-width is naturally related to clique-width and N L C -width of graphs. Graphs with bounded threshold-width generalize threshold graphs and difference graphs and extend their properties. We characterize graphs with bounded threshold-width and study in detail graphs with threshold-width at most 2 .