Whenever $G=(V,E)$ is a DAG, we can look at the induced bipartite graph and find a maximum matching. Then the number of unmatched vertices in each part of the bipartition is supposedly exactly the same as the number of paths in a minimal path cover for $G$. We proved this for the special case when $G$ is a tree, or exactly 1 edge more than a tree. However, we can't seem to find a proof for the general case.
2026-03-29 15:03:15.1774796595
DAG bipartite matchings vs. minimal path cover
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