We give a unified approach to analysing, for each positive integer s, a
class of finite connected graphs that contains all the distance transitive
graphs as well as the locally s-arc transitive graphs of diameter at least
s. A graph is in the class if it is connected and if, for each vertex v,
the subgroup of automorphisms fixing v acts transitively on the set of
vertices at distance i from v, for each i from 1 to s. We prove that
this class is closed under forming normal quotients. Several graphs in the
class are designated as degenerate, and a nondegenerate graph in the class is
called basic if all its nontrivial normal quotients are degenerate. We prove
that, for s≥2, a nondegenerate, nonbasic graph in the class is either a
complete multipartite graph, or a normal cover of a basic graph. We prove
further that, apart from the complete bipartite graphs, each basic graph admits
a faithful quasiprimitive action on each of its (1 or 2) vertex orbits, or a
biquasiprimitive action. These results invite detailed additional analysis of
the basic graphs using the theory of quasiprimitive permutation groups.Comment: Revised after referee report
