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 Wohrle and Thomas. Here, we recall the definition of the VRSP and the two modified graph-decompositions and introduce three new graph-decomposition theorems. The first new theorem decomposes a graph with respect to the semicomplete bipartite subgraphs of the graph. For the second new theorem, we introduce a matrix graph, which is used to decompose a graph in a manner similar to the decomposition of graphs using the Cartesian product. In the third new theorem, we combine these two types of decomposition. Ultimately, the goal of these graph-decomposition theorems is to come to a prime-graph decomposition.
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With summaries in Dutch, Esperanto and English. DOI: 10.4233/uuid:d7132920-346e-47c6-b754-00dc5672b437 "The subject of this study is deformation analysis of the earth's surface (or part of it) and spatial objects on, above or below it. Such analyses are needed in many domains of society. Geodetic deformation analysis uses various types of geodetic measurements to substantiate statements about changes in geometric positions.Professional practice, e.g. in the Netherlands, regularly applies methods for geodetic deformation analysis that have shortcomings, e.g. because the methods apply substandard analysis models or defective testing methods. These shortcomings hamper communication about the results of deformation analyses with the various parties involved. To improve communication solid analysis models and a common language have to be used, which requires standardisation.Operational demands for geodetic deformation analysis are the reason to formulate in this study seven characteristic elements that a solid analysis model needs to possess. Such a model can handle time series of several epochs. It analyses only size and form, not position and orientation of the reference system; and datum points may be under influence of deformation. The geodetic and physical models are combined in one adjustment model. Full use is made of available stochastic information. Statistical testing and computation of minimal detectable deformations is incorporated. Solution methods can handle rank deficient matrices (both model matrix and cofactor matrix). And, finally, a search for the best hypothesis/model is implemented. Because a geodetic deformation analysis model with all seven elements does not exist, this study develops such a model.For effective standardisation geodetic deformation analysis models need: practical key performance indicators; a clear procedure for using the model; and the possibility to graphically visualise the estimated deformations."
