  Recently, we have introduced and modified two graph-decomposition theorems
based on a new graph product, motivated by applications in the context of
synchronising periodic real-time processes. This vertex-removing synchronised
product (VRSP), is based on modifications of the well-known Cartesian product,
and is closely related to the synchronised product due to Wohrle and Thomas.
Here, we recall the definition of the VRSP and the two modified
graph-decompositions and introduce three new graph-decomposition theorems. The
first new theorem decomposes a graph with respect to the semicomplete bipartite
subgraphs of the graph. For the second new theorem, we introduce a matrix
graph, which is used to decompose a graph in a manner similar to the
decomposition of graphs using the Cartesian product. In the third new theorem,
we combine these two types of decomposition. Ultimately, the goal of these
graph-decomposition theorems is to come to a prime-graph decomposition.
