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Recently, we have introduced and modified two graph-decomposition theoremsbased on a new graph product, motivated by applications in the context ofsynchronising periodic real-time processes. This vertex-removing synchronisedproduct (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 modifiedgraph-decompositions and introduce three new graph-decomposition theorems. Thefirst new theorem decomposes a graph with respect to the semicomplete bipartitesubgraphs of the graph. For the second new theorem, we introduce a matrixgraph, which is used to decompose a graph in a manner similar to thedecomposition of graphs using the Cartesian product. In the third new theorem,we combine these two types of decomposition. Ultimately, the goal of thesegraph-decomposition theorems is to come to a prime-graph decomposition.

Three new decompositions of graphs based on a vertex-removing synchronised graph product