Are strong monomorphisms coretrations in the category of graphs?

Question

Consider the category of digraphs with strong homomorphisms as the morphisms. Here a strong homomorphism $f: G\longrightarrow H$, is a graph homomorphism which preserves and reflects adjacency of vertices in $G$ and $H$. It is true that strong monomorphisms(resp. epimorphisms) are injective (resp. surjective) in this category. Is it also true that every strong monomorphism(resp. epimorphisms) is a coretration (resp. retraction) in this category?

Answer 1

Here's monomorphism which is not a coretraction (I like the alternate term "section" coming from geometry). Let $G$ a graph with one vertex called "$0$" and no edges. Let $H$ be the graph with the two vertices $\{0,1\}$ and an edge between them. Let $f: G \to H$ be the monomorphism sending $0 \mapsto 0$. Then there is no retraction of $f$ because there is nowhere for it to send the edge.

Here's an epimorphism which is not a retraction. Let $H$ be as above, and let $K$ be a graph with one vertex called "$0$" and one loop at that vertex. Let $g: H \to K$ be the unique homomorphism. Then $g$ has no section because there's nowhere to send the loop.

The first example, but not the second, would be fixed by working with reflexive graphs, i.e. graphs where every vertex has a distinguished loop and the homomorphisms are required to preserve the distinguished loop. But here's another example of a non-retract monomorphism that works for reflexive graphs too. Let $G$ be as above, and let $L$ be the graph with 3 vertices $\{0,1/2,1\}$, and an edge from $0$ to $1/2$ and from $1/2$ to $1$. Then the obvious inclusion from $G$ to $L$ has no retraction.