In principle, it is possible to construct a graphitic P-surface with neighbored octagons separated by any Goldberg vector, (h, k). Most of the beaded P-surfaces I posted before have the vector (2,0). The smallest graphitic P-surface has a vector of (h,k)=(1,0) and its bead model was built by Chern long time ago.
Here is the bead model of P-surface with (1,1) I made this weekend. Unlike (2,0)-surface, the bead model of (1,1)-surface is a little bit crowded though.
Single unit cell: