  Recently, we have introduced 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 recall
the definition of the VRSP and the two graph-decomposition theorems, we relax
the requirements of these two graph-decomposition theorems and prove these two
(relaxed) graph-decomposition theorems.
