Connectivity of $\tau$-tilting graphs for quasi-tilted algebras and quotients of $g$-tame algebras

This note investigates the connectivity of $\tau$-tilting graphs for algebras from the point of view of quotients. We establish the connectivity of $\tau$-tilting graph for an arbitrary quasi-tilted algebra and prove that the connectivity of the $\tau$-tilting graph of a $g$-tame algebra is preserved under quotient. In particular, quotient algebras of skew-gentle algebras and quotient algebras of tame hereditary algebras have connected $\tau$-tilting graphs.