Bongartz Completion via c-Vectors.

In the present paper, we first give a characterization for Bongartz completion in |$\tau $| -tilting theory via |$c$| -vectors. Motivated by this characterization, we give the definition of Bongartz completion in cluster algebras using |$c$| -vectors. Then we prove the existence and uniqueness of Bongartz completion in cluster algebras. We also prove that Bongartz completion admits certain commutativity. We give two applications for Bongartz completion in cluster algebras. As the first application, we prove the full subquiver of the exchange quiver (or known as oriented exchange graph) of a cluster algebra |$\mathcal A$| whose vertices consist of the seeds of |$\mathcal A$| containing particular cluster variables is isomorphic to the exchange quiver of another cluster algebra. As the second application, we prove that in a |$Y$| -pattern over a universal semifield, each |$Y$| -seed (up to a |$Y$| -seed equivalence) is uniquely determined by the negative |$y$| -variables in this |$Y$| -seed. [ABSTRACT FROM AUTHOR]