Flip-graphs of non-orientable filling surfaces

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