Automorphisms of trivalent graphs

Let G g , b be the set of all uni/trivalent graphs representing the combinatorial structures of pant decompositions of the oriented surface Σ g , b of genus g with b boundary components. We describe the set A g , b of all automorphisms of graphs in G g , b showing that, up to suitable moves changing the graph within G g , b , any such automorphism can be reduced to elementary switches of adjacent edges.