  Important objects of study in $\tau$-tilting theory include the
$\tau$-tilting pairs over an algebra on the form $kQ/I$, with $kQ$ being a path
algebra and $I$ an admissible ideal. In this paper, we study aspects of the
combinatorics of mutation quivers of support $\tau$-tilting pairs, simply
called mutation quivers. In particular, we are interested in identifying
connected components of the underlying graphs of such quivers. We give a class
of algebras with two simple modules such that every algebra in the class has at
most two connected components in its mutation quiver, generalizing a result by
Demonet, Iyama and Jasso (2017). We also give examples of algebras with
strictly more than two components in their mutation quivers.
