is obtained from $B_r(x)$ in this way.

By construction, every vertex of $H(x)$ is at distance less than $g(k)/4$ from $x$; otherwise such a vertex could have been added to $X$, contradicting its maximality. Hence the diameter of $H(x)$ is less than $g(k)/2$ and so by Lemma \[lem:diameter\], $H(x)$ contains no long cycle. This in turn implies that every block of $H(x)$ has diameter at most $(\ell-1)/2$.\
We now analyse the block-tree structure of $H(x)$. First, we recall and introduce a few definitions. A *block* is a maximal connected subgraph without a cut-vertex; in particular, an edge can be a block. A *leaf-block* of a graph $H$ is a block of $H$ that contains at most one cut-vertex of $H$. Given a block $L$ of $H(x)$, an *ancestor-block of $L$* is a block of $H(x)$ that is distinct from $L$ and which has at least two vertices in common with a shortest path between $V(L)$ and $x$.

For a correct analysis, we will also need to consider a subgraph $H^*(x)$ which consists of those blocks that are ‘not too close to the boundary of $H(x)$’. More formally, we define $H^*(x)$ to be the graph induced by the union of all 