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Structure of Graphs with Locally Restricted Crossings

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Vida Dujmović, David Eppstein, David R. Wood

Combinatorics · Submitted 14 Jun 2015

abstract compiled · 2026-09-23 · no language-model review

The authors' main claim We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor.

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arXiv ID1506.04380 (version 2)
Pith IDGR5DT4NJ, Pith's permanent record of this paper and its review
CitationDavid Eppstein, David R. Wood, Vida Dujmovi\'c. "Structure of Graphs with Locally Restricted Crossings." arXiv:1506.04380 (2015). https://pith.science/paper/GR5DT4NJ
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The compiled record

The paper's own sentences, arranged. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

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Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,587 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.24 · human participants 0.01 · controlled comparison 0.06 · randomized assignment 0.01 · states quantitative result 0.98 · compares with prior results 0.96 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; .

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ treewidth bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for $g<m$, every $m$-edge graph can be embedded on a surface of genus~$g$ with $O((m/(g+1))\log^2 g)$ crossings per edge, which is tight to a polylogarithmic factor.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. background 0.76 0.37 0.04 unclear 0.81
s1 We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. result 0.99 0.97 0.81 asserted 1.00
s2 As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ treewidth bound. result 0.97 0.90 0.20 asserted 1.00
s3 Analogous results are proved for map graphs defined with respect to any surface. result 0.88 0.89 0.04 asserted 0.99
s4 Finally, we show that for $g<m$, every $m$-edge graph can be embedded on a surface of genus~$g$ with $O((m/(g+1))\log^2 g)$ crossings per edge, which is tight to a polylogarithmic factor. result 0.98 0.94 0.09 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

What was examined, and what was not

sources read
  • arXiv listing abstract v2
not examined
  • full text
  • figures and tables
  • references and citing sentences
  • relations to other papers
levels run
  • P abstract
not run
  • S sections
  • T body sentences
  • C citations
  • R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

The paper
Original abstract

We consider relations between the size, treewidth, and local crossing number (maximum number of crossings per edge) of graphs embedded on topological surfaces. We show that an $n$-vertex graph embedded on a surface of genus $g$ with at most $k$ crossings per edge has treewidth $O(\sqrt{(g+1)(k+1)n})$ and layered treewidth $O((g+1)k)$, and that these bounds are tight up to a constant factor. As a special case, the $k$-planar graphs with $n$ vertices have treewidth $O(\sqrt{(k+1)n})$ and layered treewidth $O(k+1)$, which are tight bounds that improve a previously known $O((k+1)^{3/4}n^{1/2})$ treewidth bound. Analogous results are proved for map graphs defined with respect to any surface. Finally, we show that for $g<m$, every $m$-edge graph can be embedded on a surface of genus~$g$ with $O((m/(g+1))\log^2 g)$ crossings per edge, which is tight to a polylogarithmic factor.

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