of the graph is one. Core As another component of the Dulmage–Mendelsohn decomposition, Dulmage and Mendelsohn defined the core of a graph to be the union of its maximum matchings. However, this concept should be distinguished from the core in the sense of graph homomorphisms, and from the k-core formed by the removal of low-degree vertices. Applications This decomposition has been used to partition meshes in finite element analysis, and to determine specified, underspecified and overspecified equations in systems of nonlinear equations. It was also used for an algorithm for rank-maximal matching. Asymmetric variant In there is a different decomposition of a bipartite graph, which is asymmetric - it distinguishes between vertices in one side of the graph and the vertices on the other side. It can be used to find a maximum-cardinality envy-free matching in an unweighted bipartite graph, as well as a minimum-cost maximum-cardinality matching in a weighted bipartite graph. References External links A good explanation of its application to systems of nonlinear equations is available in this paper: An open source implementation of the algorithm is available as a part of the sparse-matrix library: SPOOLES Graph-theoretical aspects of constraint solving in the SST project: Graph algorithms Matching (graph theory)