Flip-graphs of non-orientable filling surfaces

Consider a surface $\Sigma$ with punctures that serve as marked points and at least one marked point on each boundary component. We build a filling surface $\Sigma_n$ by singling out one of the boundary components and denoting by $n$ the number of marked points it contains. We consider the triangulations of $\Sigma_n$ whose vertices are the marked points and the associated flip-graph $\mathcal{F}(\Sigma_n)$. Quotienting $\mathcal{F}(\Sigma_n)$ by the homeomorphisms of $\Sigma$ that fix the privileged boundary component results in a finite graph $\mathcal{MF}(\Sigma_n)$. Bounds on the diameter of $\mathcal{MF}(\Sigma_n)$ are available when $\Sigma$ is orientable and we provide corresponding bounds when $\Sigma$ is non-orientable. We show that the diameter of this graph grows at least like $5n/2$ and at most like $4n$ as $n$ goes to infinity. If $\Sigma$ is an unpunctured M\"obius strip, $\mathcal{MF}(\Sigma_n)$ coincides with $\mathcal{F}(\Sigma_n)$ and we prove that the diameter of this graph grows exactly like $5n/2$ as $n$ goes to infinity.