We identify all $[1, λ, 8]$-cycle regular $I$-graphs and all $[1, λ, 8]$-cycle regular double generalized Petersen graphs. As a consequence we describe linear recognition algorithms for these graph families. Using structural properties of folded cubes we devise a $o(N \log N)$ recognition algorithm for them. We also study their $[1,λ,4]$, $[1,λ,6]$ and $[2, λ, 6]$-cycle regularity and settle the value of parameter $λ$.