  In conventional site percolation, all lattice sites are occupied with the
same probability. For a bipartite lattice, sublattice-selective percolation
instead involves two independent occupation probabilities, depending on the
sublattice to which a given site belongs. Here, we determine the corresponding
phase diagram for the two-dimensional square and Lieb lattices from quantifying
the parameter regime where a percolating cluster persists for
sublattice-selective percolation. For this purpose, we present an adapted
Newman-Ziff algorithm. We also consider the critical exponents at the
percolation transition, confirming previous Monte Carlo and
renormalization-group findings that suggest sublattice-selective percolation to
belong to the same universality class as conventional site percolation. To
further strengthen this conclusion, we finally treat sublattice-selective
percolation on the Bethe lattice (infinite Cayley tree) by an exact solution.
