Automorphisms of trivalent graphs

Let Gg,bGg,b be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface Σg,bΣg,b of genus gg with bb boundary components. We describe the set Ag,bAg,b of all automorphisms of graphs in Gg,bGg,b showing that, up to suitable moves changing the graph within Gg,bGg,b, any such automorphism can be reduced to elementary switches of adjacent edges.