  Let $\Gamma$ be a connected $G$-vertex-transitive graph, let $v$ be a vertex
of $\Gamma$ and let $L=G_v^{\Gamma(v)}$ be the permutation group induced by the
action of the vertex-stabiliser $G_v$ on the neighbourhood $\Gamma(v)$. Then
$(\Gamma,G)$ is said to be \emph{locally-$L$}. A transitive permutation group
$L$ is \emph{graph-restrictive} if there exists a constant $c(L)$ such that,
for every locally-$L$ pair $(\Gamma,G)$ and an arc $(u,v)$ of $\Gamma$, the
inequality $|G_{uv}|\leq c(L)$ holds.
  Using this terminology, the Weiss Conjecture says that primitive groups are
graph-restrictive. We propose a very strong generalisation of this conjecture:
a group is graph-restrictive if and only if it is semiprimitive. (A transitive
permutation group is said to be \emph{semiprimitive} if each of its normal
subgroups is either transitive or semiregular.) Our main result is a proof of
one of the two implications of this conjecture, namely that graph-restrictive
groups are semiprimitive. We also collect the known results and prove some new
ones regarding the other implication.
