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We explore several families of flip-graphs, all related to polygons orpunctured polygons. In particular, we consider the topological flip-graphs ofonce-punctured polygons which, in turn, contain all possible geometricflip-graphs of polygons with a marked point as embedded sub-graphs. Our mainfocus is on the geometric properties of these graphs and how they relate to oneanother. In particular, we show that the embeddings between them are stronglyconvex (or, said otherwise, totally geodesic). We also find bounds on thediameters of these graphs, sometimes using the strongly convex embeddings.Finally, we show how these graphs relate to different polytopes, namely type Dassociahedra and a family of secondary polytopes which we call pointihedra.

Once punctured disks, non-convex polygons, and pointihedra