diff --git a/arxiv_cbd.tex b/arxiv_cbd.tex index 5f270e98135447f9beb848e238241edf681dcf6f..83e5749ad205df59f124e2d8e073f7e4dd3fdf98 100644 --- a/arxiv_cbd.tex +++ b/arxiv_cbd.tex @@ -778,7 +778,7 @@ Hence, the condition (c1) holds. Consider now a vertex $v_i$ and a clique $C$. As we observed before, if $v_i\not\in V(C)$, then $p(C) \not\in h(v_i)$, and thus $h^\varepsilon(C)$ and $h(v_i)$ are disjoint (for sufficiently small $\varepsilon>0$). If $v_i\in C$, then the definitions ensure that $p(C)[i]$ is equal to the maximum of $h(v_i)[i]$, -and that for $j\neq i$, $p(C)[j]$ is in the interior of $h(v_i)[j]$, implying +and that for $j\neq i$, $p(C)[j]$ is in $h(v_i)[j]$, implying that $h(v_i)[j] \cap h^\varepsilon(C)[j] = [p(C)[j],p(C)[j]+\varepsilon]$ for sufficiently small $\varepsilon>0$. \end{proof} The \emph{treewidth} $\tw(G)$ of a graph $G$ is the minimum $k$ such that $G$ is a subgraph of a $k$-tree.