Graph Epimorphism

Question

What does Graph epimorphism mean? Does that implies a surjective homomorphism (A mapping between graphs that preserves their structure) of graphs? Can you please explain it?

Answer 1

An epimorphism from G to H is a surjective function $f : V(G) → V(H)$ such that

• $\forall$ $u,v ∈ V(G)$ ,if $(u,v)∈A(G)$,then $(f(u),f(v))∈A(H)$ (graph homomorphism)

• $\forall$ $(u′, v′) ∈ A(H)$, there exists $(u, v) ∈ A(G)$ such that $f (u) = u′$ and $f (v) = v′$ (surjectivity on arcs) i.e., every node in H is an image of some node in G. .

Hence we call $f$ an $Epimorphism$ if every node or arc in $H$ is image of a vertex or arc in $G$.