  Recently Beaton, de Gier and Guttmann proved a conjecture of Batchelor and
Yung that the critical fugacity of self-avoiding walks interacting with
(alternate) sites on the surface of the honeycomb lattice is $1+\sqrt{2}$. A
key identity used in that proof depends on the existence of a parafermionic
observable for self-avoiding walks interacting with a surface on the honeycomb
lattice. Despite the absence of a corresponding observable for SAW on the
square and triangular lattices, we show that in the limit of large lattices,
some of the consequences observed for the honeycomb lattice persist
irrespective of lattice. This permits the accurate estimation of the critical
fugacity for the corresponding problem for the square and triangular lattices.
We consider both edge and site weighting, and results of unprecedented
precision are achieved. We also \emph{prove} the corresponding result fo the
edge-weighted case for the honeycomb lattice.
