树分解中展开度与宽度之间的权衡
Trade-off between spread and width for tree decompositions
AI总结：
研究树分解中展开度与宽度的权衡，证明展开度与宽度之间存在理论界限，并展示近最优展开度与宽度的同步实现。
摘要：
我们研究了树分解中（平均）展开度与宽度之间的权衡，回答了Wood [arXiv:2509.01140]提出的一些问题。一个顶点v在树分解中的展开度是包含v的袋的数量。Wood询问对于哪种c>0，存在c'使得每个图G都有一个树分解，其宽度为c·tw(G)，且每个顶点v的展开度至多为c'(d(v)+1)。我们证明c≥2是必要的，而c>3是充分的。此外，我们完全回答了第二个问题，证明近最优的平均展开度可以同时实现宽度O(tw(G))。
英文摘要：
The spread of a vertex $v$ in a tree decomposition is the number of bags that contain $v$. We study the trade-off between spread and width in tree decompositions, answering every open question from Wood [arXiv:2509.01140]. First, Wood asked for the infimum of $c > 0$ such that 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 the answer is $3$. Second, we prove a conjecture of Wood, stating that every tree-decomposition of the $(n \times n)$-grid with width $n$ has a vertex with spread $Ω(n)$. Finally, we answer the last question of Wood by showing that near-optimal average spread can be achieved simultaneously with width $O(tw(G))$.
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