to critical exponents which are valid above the upper critical dimension, which is $d_c=6$ for percolation.

In this paper, we consider the removal of a slightly more complicated pattern: we remove at each step not only the leaf but also its neighbor (and consequently all adjacent edges). To avoid cumbersome circumlocutions, in the rest of this paper, we call *leaf* the pair “standard leaf + its neighbor”. Now leaf removal deletes two vertices (a vertex with a single neighbor and this neighbor) and all the edges adjacent to one or both vertices. It is quite natural to study the removal of these patterns because it has a number of applications to graph theory: several numerical characteristics of a graph behave nicely under leaf removal. One such characteristic, which was our original motivation from physics, is the multiplicity of the eigenvalue $0$ in the adjacency matrix of the graph. Others are the minimal size of a vertex cover and the maximal size of an edge disjoint subset (the matching problem), questions which are related to various combinatorial optimization problems.

The matching problem had already led mathematicians to a thorough study of leaf removal (see Refs. [@karpsipser; @pitteletal] and references therein). In 