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The VC-Dimension of Graphs with Respect to k-Connected Subgraphs

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Andrea Munaro

Discrete Mathematics · Submitted 26 Feb 2013

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

The authors' main claim In particular, we give tight upper and lower bounds for the VC-dimension.

how it is stated · asserted 1.00 read the record →

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arXiv ID1302.6500 (version 2)
Pith IDPSGJBNOR, Pith's permanent record of this paper and its review
CitationAndrea Munaro. "The VC-Dimension of Graphs with Respect to k-Connected Subgraphs." arXiv:1302.6500 (2013). https://pith.science/paper/PSGJBNOR
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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, 3,636 input tokens, $0.0002 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.80 · human participants 0.01 · controlled comparison 0.01 · randomized assignment 0.01 · states quantitative result 0.74 · compares with prior results 0.09 · mentions code or data availability 0.02.

The abstract, sentence by sentence

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

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width.

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 study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. contribution 0.90 0.76 0.08 asserted 0.86
s1 In particular, we give tight upper and lower bounds for the VC-dimension. result 0.85 0.95 0.48 asserted 1.00
s2 Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. result 0.97 0.94 0.29 asserted 1.00
s3 On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width. result 0.98 0.92 0.10 asserted 0.98

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

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  • references and citing sentences
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  • P abstract
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  • 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 study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its $k$-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is $\mathsf{NP}$-complete and that it remains $\mathsf{NP}$-complete for split graphs and for some subclasses of planar bipartite graphs in the cases $k = 1$ and $k = 2$. On the positive side, we observe it can be decided in linear time for graphs of bounded clique-width.

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