  Let $\Omega$ be a set of cardinality $n$, $G$ a permutation group on
$\Omega$, and $f:\Omega\to\Omega$ a map which is not a permutation. We say that
$G$ synchronizes $f$ if the semigroup $\langle G,f\rangle$ contains a constant
map.
  The first author has conjectured that a primitive group synchronizes any map
whose kernel is non-uniform. Rystsov proved one instance of this conjecture,
namely, degree $n$ primitive groups synchronize maps of rank $n-1$ (thus, maps
with kernel type $(2,1,\ldots,1)$). We prove some extensions of Rystsov's
result, including this: a primitive group synchronizes every map whose kernel
type is $(k,1,\ldots,1)$. Incidentally this result provides a new
characterization of imprimitive groups. We also prove that the conjecture above
holds for maps of extreme ranks, that is, ranks 3, 4 and $n-2$.
  These proofs use a graph-theoretic technique due to the second author: a
transformation semigroup fails to contain a constant map if and only if it is
contained in the endomorphism semigroup of a non-null (simple undircted) graph.
  The paper finishes with a number of open problems, whose solutions will
certainly require very delicate graph theoretical considerations.
