Let $N_{k,r}$ be a non-orientable surface of genus $k$ (i.e the connected sum of $k$ projective planes) and with $p\geq 1$ punctures.

I'm looking at ideal triangulations of the surface, namely maximal collection of pairwise disjoint ideal arcs (joining punctures) on the surface.

Given a triangulation and an arc $\alpha$ in that triangulation, we can perform a flip along $\alpha$ : the arc belongs to two triangles forming a quadrilateral, the flip along $\alpha$ gives the new triangulation where the arc has been replaced by the other diagonal of the quadrilateral, and all the other arcs remains unchanged. (When the arc belongs to only one triangle, we cannot do the flip.)

A classical result in the orientable case states that given any two triangulations of an orientable surface, there is a sequence of flips that transform one into the other (Hatcher, *On triangulations of surfaces*). So the so-called triangulation graph, whose vertices are triangulations and edges are flips, is connected.

I am looking for the similar result for non-orientable surfaces. Does anyone have a reference ? My guess is that the result should also hold but I did not find anything on the litterature.