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We construct a functor from the category of graphs to the category of groupswhich is faithful and "almost" full, in the sense that it induces bijections ofthe Hom sets up to trivial homomorphisms and conjugation in the category ofgroups.  We provide several applications of this construction to localizations (i.e.idempotent functors) in the category of groups and the homotopy category.

An "almost" full embedding of the category of graphs into the category of groups