  $2$-stratifolds are a generalization of $2$-manifolds that occur as objects
in applications such as in TDA. These spaces can be described by an associated
bicoloured labelled graph. In previous papers we obtained a classification of
1-connected trivalent $2$-stratifolds. In this paper we classify trivalent
$2$-stratifolds that have fundamental group $\mathbb{Z}$. This classification
implicitly gives an efficient algorithm that applies to a bicoloured labelled
graph to decide wether the associated $2$-stratifold has fundamental group
$\mathbb{Z}$.
