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A perfect matching is a matching of all the vertices of a graph. In other words, a perfect matching is a set of edges such that each vertex of the graph is incident to exactly one edge in the set.
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Hopcroft–Karp Algorithm for a dynamic graph
As so you all know, we have Hopcroft–Karp Algorithm for maximum matching between two sides in a bipartite graph. It runs in $O(\sqrt{V} \times E)$ where $V$ is the vertex set and $E$ is the edges set. …