We provide a technique to find a cluster-tilting object having a given cluster-tilted algebra as endomorphism ring in the finite type case. The distribution of cluster-tilting objects of type D nIn this section we will show how to, given the quiver Q of a cluster-tilted algebra of type D n , explicitly find a cluster-tilting object in the cluster category of type D n inducing it. The goal is to prove the following theorem:Theorem 4.1. Given the quiver Q of a cluster-tilted algebra of type D, it will be of one of the types 1) ⋆ ⋆ 2) ⋆ ⋆ 3)

We derive a method for mutating quivers of 2-CY tilted algebras that have loops and 2-cycles, under certain specific conditions. Further, we give the classification of the 2-CY tilted algebras coming from standard algebraic 2-CY triangulated categories with a finite number of indecomposables. These algebras satisfy the setup for our method of mutation.

Any cluster-tilted algebra is the relation extension of a tilted algebra. We present a method to, given the distribution of a cluster-tilting object in the Auslander-Reiten quiver of the cluster category, construct all tilted algebras whose relation extension is the endomorphism ring of this cluster-tilting object.Comment: Section 3 has been removed and now is an independent article (arXiv:0912.2911v1). Section 1 and 2.2 have been modified to cope with the removal of section 3. Proof of theorem 3.5 (previously 4.5) has been improved. More details have been added to section 6 (previously 7) to clarify how section 3 (previously 4) generalizes to the infinite cas

a b s t r a c tWe present a graded mutation rule for quivers of cluster-tilted algebras. Furthermore, we give a technique for recovering a cluster-tilting object from its graded quiver in the cluster category of coh X.

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