Automorphisms of trivalent graphs

TL;DR

This paper characterizes the automorphisms of uni/trivalent graphs representing pant decompositions of surfaces, showing they can be simplified to elementary edge switches through specific moves.

Contribution

It provides a classification of automorphisms of these graphs, demonstrating they are reducible to elementary switches within the graph set.

Findings

Automorphisms can be reduced to elementary edge switches.

The classification applies to graphs representing surface decompositions.

Moves within the graph set simplify automorphism analysis.

Abstract

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.