Classifying a family of symmetric graphs

Let Γ be a G-symmetric graph admitting a nontrivial G-invariant partition ℬ of block size υ. For blocks B, C of ℬ adjacent in the quotient graph Γℬ, let k be the number of vertices in B adjacent to at least one vertex in C. In this paper we classify all possibilities for (Γ Γℬ, G) in the case where k = υ − 1 ≥ 2 and ℬ(α) = ℬ(β) for adjacent vertices α β of Γ where for a vertex of Γ, say γ ∈ B, ℬ(γ) denotes the set of blocks C such that γ is the only vertex in B not adjacent to any vertex in C.