Double Occurrence Words with the Same Alternance Graph

Let \(m\) be a double occurrence word (i.e., each letter occurring in \(m\) occurs precisely twice). An alternance of \(m\) is a non-ordered pair \(uw\) of distinct letters such that we meet alternatively \(\dots v \dots w \dots v \dots w \dots\) when reading \(m\). The alternance graph \(A(m)\) is the simple graph whose vertices are the letters of \(m\) and whose edges are the alternances of \(m\). We define a transformation of double occurrence words such that whenever \(A(m) = A(n)\), \(m\) and \(n\) are related by a sequence of these transformations.