An approach to the structure of imprimitive graphs which are not multicovers

Denote by G the set of triples (Γ, X,B), where Γ is a finite X-symmetric graph of valency val(Γ) ≥ 1, B is a nontrivial X-invariant partition of V (Γ) such that ΓB, the quotient graph of Γ with respect to B, is nonempty and Γ is not a multicover of ΓB. In this article, for any given X-symmetric graph Σ, we aim to give a sufficient and necessary condition for the existence of (Γ, X,B) ∈ G, such that ΓB ∼= Σ and Γ is (X, s)-arc-transitive for some integer s ≥ 1. And in this condition, a practicable method will be given to construct such an (X, s)-arc-transitive graph Γ. Moreover, for any (Γ, X,B) ∈ G, we will show that there exists a finite integer m ≥ 1, such that after m times of such constructing process, we can obtain an X-symmetric graph ΓBm from ΓB, such that either Γ ∼= ΓBm or Γ is a multicover of ΓBm.