leaf removal process is called an history.

**Isolated points, $I$; Core of a graph, $C$**: The last term in an history starting from $G$ is a graph which splits into a collection of isolated points $I$, and an induced subgraph $C$ of $G$ without leaves or isolated points which we call the core of $G$. We denote the number of points in the core by $N_c$ and the number of edges in the core by $L_c$.

For these definitions to make sense, one has to show that the number of isolated points and the core are well defined, that is, do not depend on the choice of history. The formal proof is given in the appendix.

**Global leaf removal process**: Start from a graph $G$. If it has no leaves, stop. Else select one leaf in every bunch of leaves. Remove from the vertex set $V$ both extremities of all the selected leaves, and define $V^{(1)}$ to be the set of remaining vertices. Let $G^{(1)}$ be the subgraph of $G$ induced by $V^{(1)}$. Note that the leaves of $G^{(1)}$ (if any) are not leaves of $G$. In this operation, the vertices belonging to a bunch of $G$ that were not 