Geodesic
Diameters and geodesic properties of generalizations of the associahedron
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015).

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The $n$-dimensional associahedron is a polytope whose vertices correspond to triangulations of a convex $(n + 3)$-gon and whose edges are flips between them. It was recently shown that the diameter of this polytope is $2n - 4$ as soon as $n > 9$. We study the diameters of the graphs of relevant generalizations of the associahedron: on the one hand the generalized associahedra arising from cluster algebras, and on the other hand the graph associahedra and nestohedra. Related to the diameter, we investigate the non-leaving-face property for these polytopes, which asserts that every geodesic connecting two vertices in the graph of the polytope stays in the minimal face containing both. 
@article{DMTCS_2015_special_285_a84,
     author = {Ceballos, C. and Manneville, T. and Pilaud, V. and Pournin, L.},
     title = {Diameters and geodesic properties of generalizations of the associahedron},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015)},
     year = {2015},
     doi = {10.46298/dmtcs.2540},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2540/}
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Ceballos, C.; Manneville, T.; Pilaud, V.; Pournin, L. Diameters and geodesic properties of generalizations of the associahedron. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015), DMTCS Proceedings, 27th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2015) (2015). doi : 10.46298/dmtcs.2540. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.2540/

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