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 $\{f(v),f(w)\}\in E(H)$.


Do the notions of regular and extremal monomorphisms coincide in $\textbf{Grph}$?



Regular and extremal monomorphisms in the category of graphs
The answer is yes.


It is clear that the extremal monomorphisms $(V\_0, E\_0) \to (V\_1, E\_1)$ are precisely the *full* graph embeddings, i.e. maps $f : V\_0 \to V\_1$ such that $f$ is injective and $(a, b) \in E\_0$ if and only if $(f (a), f (b)) \in E\_1$. It is also clear that regular monomorphisms are graph embeddings, so it remains to be shown that every full graph embedding is a regular monomorphism. But this is straightforward: given a graph $(V\_1, E\_1)$ and a full subgraph $(V\_0, E\_0)$, we can easily construct a graph $(V\_2, E\_2)$ containing two isomorphic copies of $(V\_1, E\_1)$ as full subgraphs whose intersection is isomorphic to $(V\_0, E\_0)$; then $(V\_0, E\_0)$ is the equaliser of the two embeddings $(V\_1, E\_1) \to (V\_2, E\_2)$.


