in Eq. (\[eq:incndn\]) leads to $$\begin{aligned}
  i(\alpha) & = & \frac{A+B+AB}{\alpha}-1, \nonumber \\
  c(\alpha) & = & \frac{(B-A)(1-A)}{\alpha},  \\ 
  l(\alpha) & = & \frac{(B-A)^2}{2\alpha}.\nonumber \end{aligned}$$ For $\alpha \leq e$, $c(\alpha) = l(\alpha) = 0$, and the core indeed has a size $o(N)$. On the other hand, the core occupies a finite fraction $c(\alpha)$ of the vertices for $\alpha >e$. The behavior of Eq (\[eq:en\]) is responsible for this geometric transition, core percolation, at $\alpha = e$. As $c(\alpha)$ and $l(\alpha) \to 0$ when $\alpha \to e^+$, these functions are continuous but their derivatives are not: the transition is of second order. Note again that core percolation appears at $\alpha=e$, contrary to backbone percolation, which occurs at $\alpha =1$.

The fact that the core of a graph is an induced subgraph of the original graph allows to give a physicist’s argument for the uniqueness of the giant component in the core. Fix $\alpha >e$. If the core of the random graph contains two or more large connected components, there was no edge with extremities in two components in the original graph. But as the total size of the large connected components is of order $N$, the absence of such an edge 