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Let $\textbf{Grph}$ be the category whose objects are graphs $G = (V,E)$ such that $V$ is a set and $E \subseteq \mathcal{P}_2(V) := \{\{a,b\} \subseteq V: a\neq b\}$. We sometimes write $E(G)$ for $E$. The morphisms are maps $f:G\to H$ such that whenever $\{v,w\}\in E(G)$ then $\{f(v),f(w)\}\in E(H)$.

How can regular epimorphisms in $\textbf{Grph}$ be characterized?

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    $\begingroup$ I'm not sure this question is trivial; the category of simple loop-free graphs (or sets equipped with symmetric irreflexive relations) is not a topos or quasi-topos and is not a particularly nice category. $\endgroup$ – Todd Trimble May 12 '15 at 11:50
  • $\begingroup$ combinatorics.org/ojs/index.php/eljc/article/view/v15i1a1 $\endgroup$ – Steve Huntsman May 12 '15 at 12:51
  • $\begingroup$ @SteveHuntsman I don't see how the link you posted answers the question; neither of the words "regular" and "epimorphism" appear in the text. $\endgroup$ – Dominic van der Zypen May 12 '15 at 14:31
  • $\begingroup$ @DominicvanderZypen -- It doesn't answer the question: it merely reinforces Todd's comment and gives context along those lines. $\endgroup$ – Steve Huntsman May 12 '15 at 14:37
  • $\begingroup$ Sorry for misunderstanding - thanks for clarifying! $\endgroup$ – Dominic van der Zypen May 12 '15 at 14:39
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I think the paper "A canonical factorization for graph homomorphisms", Barry Fawcett, Can J. Math. 29 (4), 1977, 738-743, answers the question.

Theorem 3 states that in $\textbf{Grph}$, strict epimorphisms are the same as extremal epimorphisms, which are the same as "full epimorphisms", meaning morphisms that are surjective on vertices and edges. The paper doesn't mention regular epimorphisms, but $\textbf{Grph}$ has pullbacks, which I think means that strict epimorphisms are the same as regular epimorphisms.

It's not strictly relevant to this question, but I can't resist mentioning this related paper by the same author about epimorphisms in the category of planar graphs, whose main result will bring a smile to the face of any categorically inclined mathematician.

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