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Some new results are presented concerning the decomposition and pivoting of large sparse systems of linear equations. The paper uses graph theoretical reasoning. Starting point are some results of Rose on triangulated graphs, separation sets, and optimal ordering of sparse matrices. In the paper it is proven that a graph (and thus at the same time the matrix it represents) can be "split" ("torn") by certain vertex sets (unknowns) such that the overall number of "fill-ins" may still be optimum, although ordering is done in all components (submatrices) separately and almost independently. The results may have some significance for very large systems where they may assist in cutting down on set up time. Also some impact on the study of the possible benefits of using more than one floating point processor in parallel may be expected.