Graphs whose Eulerian trails have unique labels

Consider an undirected graph whose edges are labeled invertibly in a group. When does every Eulerian trail from one fixed vertex to another have the same label? We give a precise structural answer to this question. Essentially, we show that each ``$3$-connected part'' is labeled over a group which is isomorphic to $\mathbb{Z}_2^k$ for some $k$. We also show that the algorithmic problem admits a polynomial-time reduction to the word problem for the group.