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 Wöhrle 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.
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