Determining the minimum vertex cover in a bipartite graph from a maximum flow/matching using the residual network rather than alternating paths
The set $(X \cap T) \cup (Y \cap S)$ is the same as the one given by the alternating path algorithm. Let $L_0,L_1,\ldots$ be the subsets considered in the alternating path algorithm (in the Wikipedia page they are called $S_0,S_1,\ldots$, but that will be confusing), and let $X_i = L_i \cap X$, $Y_i = L_i \cap Y$.


A vertex $x \in X_0$ is unmatched and so the edge from $s$ to $x$ is in the residual network, showing that $x \in S$. In other words, $X_0 \subseteq S$. Analogously, $Y_0 \subseteq T$.
A vertex $x \in X_1$ is connected to some $y \in Y_0$ via some edge not in the matching. Since $(x,y)$ is not in the matching, it belongs to the residual network. Since $y \in T$, also $x \in T$. In other words, $X_1 \subseteq T$. Analogously, $Y_1 \subseteq S$.
A vertex $x \in X_2$ is connected to some $y \in Y_1$ via some edge not in the matching. As before, $(x,y)$ belongs to the residual network, and so $y \in S$ implies $x \in S$. In other words, $X_2 \subseteq S$ and $Y_2 \subseteq T$.


Continuing this way, we conclude that
$$
\begin{align*}
S \cap X &amp;= X_0 \cup X_2 \cup \cdots, &amp; T \cap X &amp;= X_1 \cup X_3 \cup \cdots, \\
S \cap Y &amp;= Y_1 \cup Y_3 \cup \cdots, &amp; T \cap Y &amp;= Y_0 \cup Y_2 \cup \cdots. \\
\end{align*}
$$
Therefore
$$
(X \cap T) \cup (Y \cap S) = X_1 \cup X_3 \cup \cdots \cup Y_1 \cup Y_3 \cup \cdots = L_1 \cup L_3 \cup \cdots,
$$
which is the output of the alternating path algorithm.