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A note on planar partial 3-trees

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arxiv 1210.8113 v1 pith:2KGPRPAS submitted 2012-10-30 cs.DM math.CO

Jan Kratochvíl , Michal Vaner This is my paper · ORCID
classification cs.DMmath.CO
keywords planarpartialtreetreesalreadyclassescolbourncompletion
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It implicitly follows from the work of [Colbourn, El-Mallah: On two dual classes of planar graphs. Discrete Mathematics 80(1): 21-40 (1990)] that every planar partial 3-tree is a subgraph of a planar 3-tree. This fact has already enabled to prove a couple of results for planar partial 3-trees by induction on the structure of the underlying planar 3-tree completion. We provide an explicit proof of this observation and strengthen it by showing that one can keep the plane drawing of the input graph unchanged.

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Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Short Paths in the Planar Graph Product Structure Theorem

    math.CO 2025-02 conditional novelty 8.0 of 10

    Every n-vertex planar graph is contained in H ⊠ P ⊠ K_c for some planar H of treewidth 3 and a path P of length O((tw(G)+1)^(1-ε) n^ε).

  2. A Gap in the 42-Queue Layout Algorithm for Planar Graphs

    cs.DM 2026-08 accept novelty 6.0 of 10

    A concrete tripod-decomposition counterexample shows that Claim 2 in the 42-queue layout paper of Bekos, Gronemann, and Raftopoulou is false, so the 42 bound is not proved.

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