ON THE ORDER OF ARC-STABILISERS IN ARC-TRANSITIVE GRAPHS, II

Let Gamma be a G-vertex-transitive graph and let (u; v) be an arc of Gamma. It is known that if the local action G(v)(Gamma)(v) (the permutation group induced by G(v) on Gamma(v)) is permutation isomorphic to the dihedral group of degree four, then either vertical bar G(uv)vertical bar is 'small' with respect to the order of Gamma or Gamma is one of a family of well-understood graphs. In this paper, we generalise this result to a wider class of local actions.