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On the K\H{o}nig-Egerv\'ary Theorem for $k$-Paths

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Stéphane Bessy, Pascal Ochem, Dieter Rautenbach

Combinatorics · Submitted 21 Oct 2017

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

The authors' main claim For $k\in \{ 3,4\}$, we give complete structural descriptions of the graphs in ${\cal G}_k$.

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arXiv ID1710.07748 (version 1)
Pith IDGSK2772Z, Pith's permanent record of this paper and its review
CitationDieter Rautenbach, Pascal Ochem, St\'ephane Bessy. "On the K\H{o}nig-Egerv\'ary Theorem for $k$-Paths." arXiv:1710.07748 (2017). https://pith.science/paper/GSK2772Z
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Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 4,471 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.35 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.11 · compares with prior results 0.64 · mentions code or data availability 0.02.

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

The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set ${\cal G}_k$ of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order $k$ equals the minimum order of a set of vertices intersecting all paths of order $k$. For $k\in \{ 3,4\}$, we give complete structural descriptions of the graphs in ${\cal G}_k$. Furthermore, for odd $k$, we give a complete structural description of the graphs in ${\cal G}_k$ that contain no cycle of order less than $k$. For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order $k$ and minimum sets of vertices intersecting all paths of order $k$.

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 The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. background 1.00 0.05 0.05 asserted 1.00
s1 Inspired by this result, we consider the set ${\cal G}_k$ of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order $k$ equals the minimum order of a set of vertices intersecting all paths of order $k$. contribution 0.86 0.38 0.10 asserted 0.76
s2 For $k\in \{ 3,4\}$, we give complete structural descriptions of the graphs in ${\cal G}_k$. result 0.51 0.96 0.35 asserted 1.00
s3 Furthermore, for odd $k$, we give a complete structural description of the graphs in ${\cal G}_k$ that contain no cycle of order less than $k$. result 0.62 0.95 0.16 asserted 1.00
s4 For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order $k$ and minimum sets of vertices intersecting all paths of order $k$. interpretation 0.44 0.94 0.22 asserted 1.00

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

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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

The famous K\H{o}nig-Egerv\'ary theorem is equivalent to the statement that the matching number equals the vertex cover number for every induced subgraph of some graph if and only if that graph is bipartite. Inspired by this result, we consider the set ${\cal G}_k$ of all graphs such that, for every induced subgraph, the maximum number of disjoint paths of order $k$ equals the minimum order of a set of vertices intersecting all paths of order $k$. For $k\in \{ 3,4\}$, we give complete structural descriptions of the graphs in ${\cal G}_k$. Furthermore, for odd $k$, we give a complete structural description of the graphs in ${\cal G}_k$ that contain no cycle of order less than $k$. For these graph classes, our results yield efficient recognition algorithms as well as efficient algorithms that determine maximum sets of disjoint paths of order $k$ and minimum sets of vertices intersecting all paths of order $k$.

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