  We study a realization of S dualities of four-dimensional $\mathcal{N}=2$
class $\mathcal{S}$ theories based on BPS graphs. S duality transformations of
the UV curve are explicitly expressed as a sequence of topological transitions
of the graph, and translated into cluster transformations of the algebra
associated to the dual BPS quiver. Our construction applies to generic class
$\mathcal{S}$ theories, including those with non-maximal flavor symmetry,
generalizing previous results based on higher triangulations. We study the the
action of S duality on UV line operators, and show that it matches precisely
with the mapping class group, by a careful analysis of framed wall-crossing. We
comment on the implications of our results for the computation of
three-manifold invariants via cluster partition functions.
