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For questions about matchings in graph theory. A matching on a graph is a set of edges such that no two edges share a common vertex.

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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. …