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The present paper shows how to construct a maximum matching in a bipartite graph with n vertices and m edges in a number of computation steps proportional to (m + n)x/.

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Submittal of an algorithm for consideration for publication in Communications of the ACM implies unrestricted use of the algorithm within a computer is permissible.Abstract: Efficient algorithms are presented for partitioning a graph into connected components, biconnected components and simple paths. The algorithm for partitioning of a graph into simple paths is iterative and each iteration produces a new path between two vertices already on paths. (The start vertex can be specified dynamically.) If V is the number of vertices and E is the number of edges, each algorithm requires time and space proportional to max (V, E) when executed on a random access computer. DescriptionGraphs arise in many different contexts where it is necessary to represent interrelations between data elements. Consequently algorithms are being developed to manipulate graphs and test them for various properties. Certain basic tasks are common to many of these algorithms. For example, in order to test a graph for planarity, one first decomposes the graph into biconnected components and tests each component separately. If one is using an algorithm [4] with asymptotic growth of V log(V) to test for planarity, it is imperative that one use an algorithm for partitioning the graph whose asymptotic growth is linear with the number of edges rather than quadratic in the number of vertices. In fact, representing a graph by a connection matrix in the above case would result in spending more time in constructing the matrix than in testing the graph for planarity if it were represented by a list of edges. It is with this in mind that we present a structure for representing graphs in a computer and several algorithms for simple

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