In Fomin, Thurston and Shapiro's Cluster algebras and triangulated surfaces. part i: Cluster complexes, a general construction is described that associates a quiver or a skew-symmetrizable matrix with a triangulation of a bordered surface. Cluster algebra is also constructed from the associated matrices. The goals of our study are:1. Determine when a graph, diagram or integer matrix is associated to a triangulation of a surface.2. Find all possible triangulations associated with a given graph, diagram or integer matrix.3. Find out when a given matrix is associated with different surfaces.In Cluster algebras and triangulated surfaces. part i: Cluster complexes and Cluster algebras of finite mutationtype via unfoldings, it is shown that the mutation type of a cluster algebra can be determined by examining its associated graph or diagram. A cluster algebra is of finite mutation type if its associated graph or diagram is obtained by gluing certain graphs under certain rules, or associated to one of finitely many exceptional diagrams. The former type of graphs and diagrams are called block-decomposable and s-decomposibable respectively. Our main resultis an algorithm that determines if a graph, diagram or matrix is block-decomposable or sdecomposable, and, hence associated with a triangulation of a surface. As a corollary, we can determine if a cluster algebra is of fine mutation type. Using the algorithm, we can also determine if a block-decomposable (or s-decomposable) exchange matrix is associated to a unique surface.The algorithm is an improvement of the one in Cluster algebras and triangulated surfaces. part i: Cluster complexes. The algorithm in the previous article is polynomial in size of matrix of power 10. Our algorithm is linear. As a possible application, our algorithm can be used in G. Musiker's Sage software package for cluster algebra computation (see .Gregg Musiker and Christian Stump. A compendium on the cluster algebra and quiver package in sage.).
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