  We present a rough estimation -- up to four significant digits, based on the
scaling hypothesis and the probability of belonging to the largest cluster vs.
the occupation probability -- of the critical occupation probabilities for the
random site percolation problem on a honeycomb lattice with complex
neighborhoods containing sites up to the fifth coordination zone. There are 31
such neighborhoods with their radius ranging from one to three and containing
from three to 24 sites. For two-dimensional regular lattices with compact
extended-range neighborhoods, in the limit of the large number $z$ of sites in
the neighborhoods, the site percolation thresholds $p_c$ follow the dependency
$p_c\propto 1/z$, as recently shown by Xun, Hao and Ziff [Physical Review E
105, 024105 (2022)]. On the contrary, noncompact neighborhoods (with holes)
destroy this dependence due to the degeneracy of the percolation threshold
(several values of $p_c$ corresponding to the same number $z$ of sites in the
neighborhoods). An example of a single-value index $\zeta=\sum_i z_i r_i$ --
where $z_i$ and $r_i$ are the number of sites and radius of the $i$-th
coordination zone, respectively -- characterizing the neighborhood and allowing
avoiding the above-mentioned degeneracy is presented. The percolation threshold
obtained follows the inverse square root dependence $p_c\propto 1/\sqrt\zeta$.
The functions boundaries() (written in C) for basic neighborhoods (for the
unique coordination zone) for the Newman and Ziff algorithm [Physical Review E
64, 016706 (2001)] are also presented.
