  Let $G_{g,b}$ be the set of all uni/trivalent graphs representing the
combinatorial structures of pant decompositions of the oriented surface 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.
