Given a graph Γ= (V, E), the graph group on Γ is the group generated by the vertex set V with the defining relations being that two adjacent vertices commute. In the following, the centralizer problem for graph groups is solved and it is shown that, for some graph groups, aut (F〈Γ〉) is finitely generated by generators which are natural analogues to the Nielsen automorphisms for a free group.
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Herman Servatius (1989) studied this question.