← 研踪 PaperTrailarXiv:2611.14327 · v1
Characterizing Fixed-Treewidth Planar Graphs
首次提交 2026-11-11 00:52
版本更新 2026-11-11 00:52
雷达观测 2026-11-11 13:07
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.
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v1 · 2026-11-11 00:52 · 最新观测版本
Characterizing Fixed-Treewidth Planar Graphs
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.
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