# Graph Epimorphism

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?

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 $$G$$.
• Yes they are sets of adjacent pairs of vertices. I assumed $G$ and $H$ are directed graphs, therefore $A$ is set of arcs(directed edges). However, we can use the word $E$ instead of $A$. – W.R.P.S Jan 14 at 18:51