On a class of finite symmetric graphs

Let ΓΓ be a GG-symmetric graph, and let BB be a nontrivial GG-invariant partition of the vertex set of ΓΓ. This paper aims to characterize (Γ,G)(Γ,G) under the conditions that the quotient graph ΓBΓB is (G,2)(G,2)-arc transitive and the induced subgraph between two adjacent blocks is 2⋅K22⋅K2 or K2,2K2,2. The results answer two questions about the relationship between ΓΓ and ΓBΓB for this class of graphs.