Then, by construction, there is a path of length at most j -1 from some v ∈ V (Γ ) to x and from v ′ ∈ V (Γ ) to y. These paths can be extended by the edge xy to paths on length at most j from v to y and v ′ to x, respectively, and thus xy ∈ E(Q j ). That is, Q j contains all edges of Υ induced by V (Q j-1 ) and only these since Q j ⊆ Υ.