If I have a bipartite graph between two sets B and G of size $N$ such that every subset $A$ of vertices in B is connected to at least $|A|$ vertices in G. If I delete all the redundant edges (those that can be removed without the condition above failing) how do I know that the new graph is a bijection between the two sets.
2026-03-25 23:15:55.1774480555
Minimal bipartite graph
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By Hall's Marriage Theorem https://en.wikipedia.org/wiki/Hall%27s_marriage_theorem there is a matching for every vertex of $B$.
Since you know that you have a matching at the start of the process then find that matching. Deleting all edges not in the matching then doesn't affect the bijection.
A suitable algorithm could be as follows.
Define a subset A of vertices in B to be critical if they are connected to |A| vertices in G. Define an edge to be critical if it is involved in such a connection.
If there are no non-critical edges then you have the minimal graph.
Otherwise, delete any non-critical edge.
Return to step 1. (New edges may have become critical after the deletion.)