COMMENTS AND REPLIES: Comment on 'Monte Carlo simulation study of the two-stage percolation transition in enhanced binary trees'

The enhanced binary tree (EBT) is a nontransitive graph which has two percolation thresholds p_c1 and p_c2 with p_c1 < p_c2. Our Monte Carlo study implies that the second threshold p_c2 is significantly lower than a recent claim by Nogawa and Hasegawa (2009 J. Phys. A: Math. Theor. 42 145001). This means that p_c2 for the EBT does not obey the duality relation for the thresholds of dual graphs p_{c2}+\overline{p}_{c1}=1 which is a property of a transitive, nonamenable, planar graph with one end. As in regular hyperbolic lattices, this relation instead becomes an inequality p_{c2}+\overline{p}_{c1}<1 . We also find that the critical behavior is well described by the scaling form previously found for regular hyperbolic lattices.