show that if Y is a subgraph of X and f is a homomorphism from X to Y such that f | Y is a bijection, then Y is a retract

Question

Show that if $Y$ is a subgraph of $X$ and $f$ is a homomorphism from $X$ to $Y$ such that $f|_Y$ is a bijection, then $Y$ is a retract.

Answer 1

It's important to take note of the logical quantifiers in reading the definition of a subgraph being a retract. A subgraph $Y \subset X$ is a retract if there exists a retraction $f : X \to Y$, where a retraction is a homomorphism $f : X \to Y$ such that $f|_{Y} = Id$.

So, even though you could find a homomorphism $f$ where $f|_{Y}$ is a bijection but not a retraction (as one commenter did), that doesn't mean a retraction does not exist.

Here's a hint: Consider that $f|_{Y}: Y \to Y$ is a bijective homomorphism. That is, $f|_{Y} \in \text{Aut}(Y)$. Can you find another graph homomorphism $g$ so that $g \circ f : X \to Y$ is a retraction?