Geodesic
A geometric model for cluster categories of type $D_n$.
Journal of Algebraic Combinatorics, Tome 27 (2008) no. 1, pp. 1-21.

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Summary: We give a geometric realization of cluster categories of type $D _{ n }$ using a polygon with $n$ vertices and one puncture in its center as a model. In this realization, the indecomposable objects of the cluster category correspond to certain homotopy classes of paths between two vertices.
Keywords: keywords cluster category, triangulated surface, punctured polygon, elementary move 
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Schiffler, Ralf. A geometric model for cluster categories of type $D_n$.. Journal of Algebraic Combinatorics, Tome 27 (2008) no. 1, pp. 1-21. http://geodesic.mathdoc.fr/item/JAC_2008__27_1_a5/