Geodesic
Maximal green sequences for arbitrary triangulations of marked surfaces (Extended Abstract)
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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In general, the existence of a maximal green sequence is not mutation invariant. In this paper we show that it is in fact mutation invariant for cluster quivers associated to most marked surfaces. We develop a procedure to find maximal green sequences for cluster quivers associated to an arbitrary triangulation of closed higher genus marked surfaces with at least two punctures. As a corollary, it follows that any triangulation of a marked surface with at least one boundary component has a maximal green sequence. 
@article{DMTCS_2020_special_379_a65,
     author = {Mills, Matthew R.},
     title = {Maximal green sequences for arbitrary triangulations of marked surfaces {(Extended} {Abstract)}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6383},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6383/}
}
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Mills, Matthew R. Maximal green sequences for arbitrary triangulations of marked surfaces (Extended Abstract). Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6383. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6383/

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