  Given any directed graph E one can construct a graph inverse semigroup G(E),
where, roughly speaking, elements correspond to paths in the graph. In this
paper we study the semigroup-theoretic structure of G(E). Specifically, we
describe the non-Rees congruences on G(E), show that the quotient of G(E) by
any Rees congruence is another graph inverse semigroup, and classify the G(E)
that have only Rees congruences. We also find the minimum possible degree of a
faithful representation by partial transformations of any countable G(E), and
we show that a homomorphism of directed graphs can be extended to a
homomorphism (that preserves zero) of the corresponding graph inverse
semigroups if and only if it is injective.
