Ture or False: for every graph G without a perfect matching and every vertex v of G there is a maximum matching M of G such that v is M-unsaturated.
2026-03-26 22:57:54.1774565874
Existence of a maximum matching such that v is un-saturated
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Consider $P_3$, the path graph on 3 nodes. It does not have a perfect matching but two maximal matchings (by picking either of the two edges present in $P_3$) neither of which leave the center vertex unsaturated.