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 pc1 and pc2 with pc1 < pc2. Our Monte Carlo study implies that the second threshold pc2 is significantly lower than a recent claim by Nogawa and Hasegawa (2009 J. Phys. A: Math. Theor. 42 145001). This means that pc2 for the EBT does not obey the duality relation for the thresholds of dual graphs pc2+p-barc1=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 pc2+p-barc1<1. We also find that the critical behavior is well described by the scaling form previously found for regular hyperbolic lattices. (comments and replies)