Let Γ be a finite G-symmetric graph whose vertex set admits a non-trivial G-invariant partition [Bscr ] with block size v. A framework for studying such graphs Γ was developed by Gardiner and Praeger which involved an analysis of the quotient graph Γ[Bscr ] relative to [Bscr ], the bipartite subgraph Γ[B, C] of Γ induced by adjacent blocks B, C of Γ[Bscr ] and a certain 1-design [Dscr ](B) induced by a block B ∈ [Bscr ]. The present paper studies the case where the size k of the blocks of [Dscr ](B) satisfies k = v − 1. In the general case, where k = v − 1 [ges ] 2, the setwise stabilizer GB is doubly transitive on B and G is faithful on [Bscr ]. We prove that [Dscr ](B) contains no repeated blocks if and only if Γ[Bscr ] is (G, 2)-arc transitive and give a method for constructing such a graph from a 2-arc transitive graph with a self-paired orbit on 3-arcs. We show further that each such graph may be constructed by this method. In particular every 3-arc transitive graph, and every 2-arc transitive graph of even valency, may occur as Γ[Bscr ] for some graph Γ with these properties. We prove further that Γ[B, C] ≅ Kv−1,v−1 if and only if Γ[Bscr ] is (G, 3)-arc transitive.