Maximum cardinality matching is a fundamental problem in graph theory. We are given a graph , and the goal is to find a matching containing as many edges as possible; that is, a maximum cardinality subset of the edges such that each vertex is adjacent to at most one edge of the subset. As each edge will cover exactly two vertices, this problem is equivalent to the task of finding a matching that covers as many vertices as possible. An important special case of the maximum cardinality matching problem is when is a bipartite graph, whose vertices are partitioned between left vertices in and right vertices in , and edges in always connect a left vertex to a right vertex. In this case, the problem can be efficiently solved with simpler algorithms than in the general case. Algorithms for bipartite graphs Flow-based algorithm The simplest way to compute a maximum cardinality matching is to follow the Ford–Fulkerson algorithm. This algorithm solves the more general problem of computing the maximum flow. A bipartite graph can be converted to a flow network as follows. Add a source vertex ; add an edge from to each vertex in . Add a sink vertex ; add an edge from each vertex in to . Assign a capacity of 1 to each edge. Since each edge in the network has integral capacity, there exists a maximum flow where all flows are integers; these integers must be either 0 or 1 since the all capacities are 1. Each integral flow defines a matching in which an edge is in the matching if and only if its flow is 1. It is a matching because: The incoming flow into each vertex in is at most 1, so the outgoing flow is at most 1 too, so at most one edge adjacent to each vertex in is present. The outgoing flow from each vertex in is at most 1, so the incoming flow is at most 1 too, so at most one edge adjacent to each vertex in is present. The Ford–Fulkerson algorithm proceeds by repeatedly finding an augmenting path from some to some and updating the matching by taking the symmetric difference of that path with (assuming such a path exists). As each path can be found in time, the running time is , and the maximum matching consists of the edges of that carry flow from