Decompositions of graphs based on a new graph product

Recently, we have introduced 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 closely related to the synchronised product due to Wöhrle and Thomas. Here, we recall the definition of the VRSP and use it to define two different decompositions of graphs. Although our main results apply to directed labelled acyclic multigraphs, the VRSP can also be used to decompose any undirected graph of order at least 4 into two smaller graphs.