  We describe a general method to determine dualities between supersymmetric 5d
gauge theories. The method is based on performing local S-dualities in the
geometry associated to the gauge theory. We find that often a duality can be
obtained by adding matter to both sides of a more primitive duality. This
allows us to define the notion of irreducible dualities which cannot be
obtained from more primitive dualities. More general dualities then are
obtained by adding matter to both sides of an irreducible duality. The
geometric method described in this paper allows us to systematically construct
irreducible dualities. As an application, we explicitly determine a special
class of irreducible dualities classified by removal and addition of edges into
a Dynkin diagram. This class of dualities vastly generalizes many of the known
5d dualities in the literature.
