Menger-decomposition of a graph and its application to the structural analysis of a large-scale system of equations

Abstract

Graph representation of a large-scale system of non-linear equations provides an efficient way of testing the structural solvability, detecting the inconsistencies in modelling and decomposing the whole system into partially ordered subsystems. In this paper, the M-decomposition is defined for a graph with specified ‘entrance’ and ‘exit’ vertices, in terms of the Menger-type linkings from the entrance to the exit. Some properties of the M-decomposition are shown; specifically it is noted that the M-decomposition agrees with the Dulmage-Mendelsohn decomposition of the associated bipartite graph. The M-decomposition is useful for the structural analysis of a large-scale system of equations; the M-decomposition leads to the finest block-triangularization and the resulting subproblems are structurally solvable. Also pointed out is the fact that among the cycles on the representation graph, only those which are contained in an M-irreducible component correspond to essential equations.