  We identify all $[1, \lambda, 8]$-cycle regular $I$-graphs and all $[1,
\lambda, 8]$-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 $[1,\lambda,4]$, $[1,\lambda,6]$ and $[2, \lambda,
6]$-cycle regularity and settle the value of parameter $\lambda$.
