that backbone percolation and standard percolation occur at the same point, namely at $\alpha=1$, has a very simple explanation for random graphs. Indeed, a large random graph of average connectivity $\alpha<1$ consists of a forest (union of finite trees) plus a finite number of finite connected components with one closed loop. Obviously, each tree shrinks to a single isolated point after leaf removal. However, at $\alpha = 1$ the percolation transition occurs and when $\alpha >1$, a random graph consists of a forest plus a finite number of components with one closed loop, plus a “giant” connected component containing a finite fraction of the vertices and an extensive number of loops. No loop is destroyed by leaf removal so that the giant component leads to a macroscopic connected remnant after leaf removal. The percolation transition was discovered and studied by Erdös and Rényi in their seminal paper [@erdos]. This has initiated a lot of work on the random graph model, and many fine details concerning the structure of the percolation transition have been computed (see e.g Ref. [@knuth]).

The random graph model is believed to be essentially equivalent to a mean field approximation for percolation on (finite dimensional) lattices, leading 