I am reading “An $n^{\frac{5}{2}}$ Algorithm for Maximum Matchings in Bipartite Graphs” by Hopcroft and Karp.
Please see the image below.
Let $s$ be the cardinality of a maximum matching.
I think any of $M_0, M_1, M_2, \cdots, M_s$ is a matching and $M_s$ is a maximum matching.
So, I think $P_s$ doesn't exist.
But the authors wrote $M_0, M_1, M_2, \cdots, M_i, \cdots$ and $|P_0|, |P_1|, \cdots, |P_i|, \cdots$.
Why?
Maybe I am confused.
