AI 中文总结
该研究证明平面图存在最优宽度且袋诱导子图路宽至多为3的树分解，还推广到排除固定双顶森林子式的图，同时给出平面图线性网格子式定理的新证明。
AI 中文摘要
我们证明，每个平面图都存在最优宽度的树分解，使得每个袋（bag）诱导的子图的路宽至多为3，该界是最优的。对于满足特定极小性条件的树分解，我们实际上给出了每个袋可能结构的精确描述。此外，我们证明任意k个袋的并的路宽为O(k)。我们还证明，排除固定双顶森林子式的图存在最优宽度的树分解，使得每个袋诱导的子图有路宽上界，这包括可嵌入任意固定曲面的图。作为该方法的副产品，我们给出了平面图线性网格子式定理的新证明。
英文摘要
We show that every planar graph has a tree-decomposition with optimal width such that the subgraph induced by each bag has pathwidth at most 3. This bound is best possible, and for tree-decompositions that satisfy a certain minimality condition, we in fact give a precise description of the possible structures in each bag. Moreover, we show that the union of any $k$ bags has pathwidth $O(k)$. We also show that graphs excluding a fixed double-apex-forest minor have a tree-decomposition with optimal width such that the subgraph induced by each bag has bounded pathwidth. This includes graphs embeddable on any fixed surface. As a byproduct of our machinery, we give a new proof of the linear grid minor theorem for planar graphs.
