  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 W\"ohrle and Thomas.
Here, we introduce a new graph-decomposition theorem based on the VRSP that
provides a Cartesian decomposition of graphs.
