Universal condition for critical percolation thresholds of kagome-like lattices

Lattices that can be represented in a kagome-like form are shown to satisfy a universal percolation criticality condition, expressed as a relation between P-3, the probability that all three vertices in the triangle connect, and P-0, the probability that none connect. A linear approximation for P-3(P-0) is derived and appears to provide a rigorous upper bound for critical thresholds. A numerically determined relation for P-3(P-0) gives thresholds for the kagome, site-bond honeycomb, (3-12(2)) lattice, and "stack-of-triangle" lattices that compare favorably with numerical results.