Forcing Brushes

Dieter Rautenbach; Dirk Meierling

RECORD

Forcing Brushes

Dirk Meierling, Dieter Rautenbach

Combinatorics · Submitted 2 Jan 2018

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

The authors' main claim We give short and simple proofs of the inequalities $B(G)\leq Z(L(G))$ and $Z(G)\leq Z(L(G))$ first established by Erzurumluo\u{g}lu, Meagher, and Pike, where $G$ is a graph without isolated vertices, $B(G)$ is the brushing number of $G$, $Z(G)$ is the zero forcing number of $G$, and $L(G)$ is the line graph of $G$.

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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-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 1,793 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.36 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.86 · compares with prior results 0.75 · mentions code or data availability 0.01.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

We give short and simple proofs of the inequalities $B(G)\leq Z(L(G))$ and $Z(G)\leq Z(L(G))$ first established by Erzurumluo\u{g}lu, Meagher, and Pike, where $G$ is a graph without isolated vertices, $B(G)$ is the brushing number of $G$, $Z(G)$ is the zero forcing number of $G$, and $L(G)$ is the line graph of $G$.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We give short and simple proofs of the inequalities $B(G)\leq Z(L(G))$ and $Z(G)\leq Z(L(G))$ first established by Erzurumluo\u{g}lu, Meagher, and Pike, where $G$ is a graph without isolated vertices, $B(G)$ is the brushing number of $G$, $Z(G)$ is the zero forcing number of $G$, and $L(G)$ is the line graph of $G$.	contribution 0.93	0.87	0.89	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 v1

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full text
figures and tables
references and citing sentences
relations to other papers

levels run

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 give short and simple proofs of the inequalities $B(G)\leq Z(L(G))$ and $Z(G)\leq Z(L(G))$ first established by Erzurumluo\u{g}lu, Meagher, and Pike, where $G$ is a graph without isolated vertices, $B(G)$ is the brushing number of $G$, $Z(G)$ is the zero forcing number of $G$, and $L(G)$ is the line graph of $G$.

Human review

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What this paper does with the arXiv papers it cites

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers whose main result relates to this one's

Found among the papers closest to this one by title and main claim. Pith's classifier compared each pair's main results and a second model family confirmed the relation; checked against two further families, shown conflicts agreed 118 times in 120 and shown same results 218 times in 239. The quoted sentence is the other paper's main claim.

States the same result 1 paper

We prove that the zero-forcing number of the line graph is an upper bound for the brushing number by constructing a brush configuration based on a zero-forcing set for the line graph.

Brushing Number and Zero-Forcing Number of Graphs and their Line Graphs · 0.85

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