固定树宽平面图的刻画
Characterizing Fixed-Treewidth Planar Graphs
Laura Merker, Samuel Schneider, Torsten Ueckerdt
AI 速读 基于摘要
作者给出对每个 k≥3 的树宽为 k 的平面图的第一个归纳刻画，并由此推出树宽至多 4 的平面图顶点荫度至多 2。
math.CO · cs.DM
查看原文摘要
We give the first inductive characterization of planar graphs of treewidth $k$ for each $k \geq 3$. Our definition generalizes the well-known inductive definition of planar $3$-trees and the recent definition of so-called planar quasi-$4$-trees [Förster, 2024]. As a key step, which may be of independent interest, we show that every planar triangulation $G$ of treewidth $k$ admits a tree decomposition of width $k$ where each adhesion set induces a cycle in $G$. As an application of planar quasi-$4$-trees, we show that planar graphs of treewidth at most $4$ have vertex arboricity at most $2$. That is, the vertex-set of every planar graph of treewidth at most $4$ can be partitioned into two sets that each induce a forest.