We study the trade-off between (average) spread and width in tree decompositions, answering several questions from Wood [arXiv:2509.01140]. The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. Wood asked for which $c>0$, there exists $c'$ such that each graph $G$ has a tree decomposition of width $c\cdot tw(G)$ in which each vertex $v$ has spread at most $c'(d(v)+1)$. We show that $c\geq 2$ is necessary and that $c>3$ is sufficient. Moreover, we answer a second question fully by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.

翻译：我们研究了树分解中（平均）扩展度与宽度之间的权衡关系，回答了Wood [arXiv:2509.01140]提出的若干问题。顶点$v$在树分解中的扩展度是指包含$v$的袋（bag）的数量。Wood提出：对于哪些$c>0$，存在常数$c'$使得任意图$G$都存在宽度为$c\cdot tw(G)$的树分解，其中每个顶点$v$的扩展度至多为$c'(d(v)+1)$？我们证明$c\geq 2$是必要条件，而$c>3$是充分条件。此外，我们通过证明在宽度为$O(tw(G))$的树分解中可同时实现接近最优的平均扩展度，从而完整回答了第二个问题。