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) + <(p)over bar>(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) + (p) over bar (c1) < 1. We also find that the critical behavior is well described by the scaling form previously found for regular hyperbolic lattices.