Analysing finite locally s-arc transitive graphs

We present a new approach to analysing finite graphs which admit a vertex intransitive group of automorphisms G and are either locally (G,s)-arc transitive for sgreater than or equal to2 or G-locally primitive. Such graphs are bipartite with the two parts of the bipartition being the orbits of G. Given a normal subgroup N which is intransitive on both parts of the bipartition, we show that taking quotients with respect to the orbits of N preserves both local primitivity and local s-arc transitivity and leads us to study graphs where G acts faithfully on both orbits and quasiprimitively on at least one. We determine the possible quasiprimitive types for G in these two cases and give new constructions of examples for each possible type. The analysis raises several open problems which are discussed in the final section.