---
author:
- |
    M. Bauer and O. Golinelli,\
    email: {bauer, golinelli}@spht.saclay.cea.fr\
    Service de Physique Théorique, Cea Saclay, 91191 Gif-sur-Yvette, France
date: |
    January 31, 2001; revised June 22, 2001.\
    Preprint T01/017; arXiv:cond-mat/0102011.\
    Eur. Phys. J. B [**24**]{}, 339–352 (2001). 
title: |
    **Core percolation in random graphs:\
    a critical phenomena analysis**
---

Introduction
============

What remains of a graph when leaves are iteratively removed until none remains ? The answer depends on what is meant by leaves.

In the most standard definition, a leaf is a vertex with exactly one neighbor, and leaf removal deletes this vertex and the adjacent edge. In the context of large random graphs where the connectivity $\alpha$ (the average number of neighbors of a vertex) is kept fixed and the number of vertices $N \to \infty$, the answer is well known and interesting. When $\alpha < 1$, the remnant after leaf removal is made of $O(N)$ isolated vertices, plus a subgraph of size $o(N)$ without leaves. When $\alpha >1$, the remnant still contains $O(N)$ isolated points, but the rest is a subgraph of size $O(N)$, which is dominated by a single connected component usually called the backbone. The a priori surprising, but rather general, fact 