The nature of the Bipartite Graph

In the bipartite graph, the number of vertices overwrites is the number of matches.

(1) Maximum number of matching points in a bipartite graph = Minimum number of points covered

That is, the minimum vertex makes each edge at least associated with one of the vertices. Obviously, you can directly obtain the maximum matching node.

(2) Independent Number of bipartite graphs = number of vertices-Maximum number of matching

Obviously, the vertices at both ends of the maximum match are removed from the vertex set. The remaining vertices are independent sets. This is | v |-2 * | M |, at the same time, it is inevitable that a vertex can be obtained from both ends of each matching edge to join the independent set and maintain its independent set nature.

(3) Minimum path overwrite of Dag = number of vertices-Maximum number of matching

After each vertex is split for the maximum match, the result is n-M. When finding a specific path, you can follow the matching edge, matching edge I → j ', J → K ', k → l '.... form a directed path.

(4) Maximum number of matches = left matching point + right unmatched point.

Because any edge in the maximum matching set is marked on the left and on the right if it is not marked on the left, we can find a new augmented path, therefore, each edge is overwritten by at least one vertex.

(5) minimum edge coverage = number of vertices in the graph-Maximum number of matching = maximum independent set.