Are there any canonical methods, in graph theory, to produce a regular graph from a non-regular one? What I'm after is a construction which is as "categorically nice" as possible; for instance, to make a graph bipartite, you can take the tensor product of $G$ and $K_2$. Even more precisely, let's call the regularized graph $f(G)$; then I'd like there to be as nice a relationship as possible between $Hom(G, -)$ and $Hom(f(G), -)$.
I'm not really expecting there to be something particularly nice that satisfies even my loose criteria, but I'd be happy to upvote/accept any useful pointers to the literature. Alternatively, if anyone has an argument explaining why such a construction shouldn't exist, that'd be helpful too.