Sub-bipartite Graph perfect matching implies Graph perfect matching?

Question

Knowing that $G$ is a Bipartite Graph $g=(S,T,E)$ and $G − {x, y}$ has a perfect matching* $\forall x\in S, \forall y\in T$, how can I demonstrate that a perfect matching for $G$ exist?

*A perfect matching is a matching which saturates all the vertices in $G$.

Answer 1

Let $x \in S$ and $y \in T$, then by the Hypothesis there is a perfect matching $M$ on $G-x,y$. Let $x*\in S-x$ and $y*\in T-y$ be two vertices connected by an edge in $M$. Then by the Hypothesis, there is a perfect Matching $M*$on $G-x*,y*$. The edge set formed by taking $M*$ and adding the edge connecting $x*$ to $y*$ in $M$ is a perfect matching for G.

Answer 2

Take any edge $xy$ in $G$.

By supposition $G-\{x,y\}$ has a perfect matching $M$.

Therefore, $M \cup \{xy\}$ is a perfect matching for $G$.