Every longest circuit of a 3‐connected, <i>K</i><sub>3,3</sub>‐minor free graph has a chord

Birmelé, Étienne

doi:10.1002/jgt.20312

Cited by 5 publications

(4 citation statements)

References 7 publications

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“…Therefore, t = 2 and hence C has length at least 8. If ℓ ≥ 6, then there exist j ∈ [2] such that {x 2 , x 4 } ⊆ N C (H j ) and {x 3 , x 5 } ⊆ N C (H 3−j ). This implies that G contains an (x, y)-path of length greater than ℓ(P ), a contradiction.…”

Section: Proof Of Theorem 12mentioning

confidence: 99%

Geography of Technology Transfer in China

Liu

2023

East China Normal University Scientific Reports

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Graphs can model complex relationships between objects, enabling a myriad of Web applications such as online page/article classification and social recommendation. While graph neural networks (GNNs) have emerged as a powerful tool for graph representation learning, in an end-to-end supervised setting, their performance heavily relies on a large amount of task-specific supervision. To reduce labeling requirement, the "pre-train, fine-tune" and "pre-train, prompt" paradigms have become increasingly common. In particular, prompting is a popular alternative to fine-tuning in natural language processing, which is designed to narrow the gap between pre-training and downstream objectives in a task-specific manner. However, existing study of prompting on graphs is still limited, lacking a universal treatment to appeal to different downstream tasks. In this paper, we propose GraphPrompt, a novel pre-training and prompting framework on graphs. GraphPrompt not only unifies pre-training and downstream tasks into a common task template, but also employs a learnable prompt to assist a downstream task in locating the most relevant knowledge from the pre-trained model in a task-specific manner. Finally, we conduct extensive experiments on five public datasets to evaluate and analyze GraphPrompt. CCS CONCEPTS• Computing methodologies → Learning latent representations; • Information systems → Data mining.

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Section: Proof Of Theorem 12mentioning

confidence: 99%

Geography of Technology Transfer in China

Liu

2023

East China Normal University Scientific Reports

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“…In particular, Zhang [11] proved that the conjecture is true for planar graphs with minimum degree at least four. More recently, Birmelé [1] verified the conjecture for the larger class of K 3,3 -minor free graphs.…”

Section: Introductionmentioning

confidence: 81%

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

Broersma

Kang

2013

Graphs and Combinatorics

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We verify two special cases of Thomassen's conjecture of 1976 stating that every longest cycle in a 3-connected graph contains a chord. We prove that Thomassen's conjecture is true for two classes of 3-connected graphs that have a bounded number of removable edges on or off a longest cycle. Here an edge e of a 3-connected graph G is said to be removable if G − e is still 3-connected or a subdivision of a 3-connected (multi)graph. We give examples to show that these classes are not covered by previous results. Keywords 3-Connected graph • Removable edge • Chord Mathematics subject classifications (2000) 05C40 • 05C38 • 05C75

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“…The first result regarding planar graphs was due to Zhang [13], who showed the conjecture holds for cubic planar graphs or planar graphs with minimum degree at least four. Subsequently, Kawarabayashi et al [5] verified the conjecture for locally 4-connected planar graphs, and Birmelé [2] verified the result for every 3-connected graph with no K 3,3 -minor. Wu et al [12] verified the result for certain classes of graphs that have a bounded number of removable edges.…”

Section: Introductionmentioning

confidence: 92%

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

Harvey

2017

Electron. J. Combin.

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A well-known conjecture of Thomassen states that every cycle of maximum order in a $3$-connected graph contains a chord. While many partial results towards this conjecture have been obtained, the conjecture itself remains unsolved. In this paper, we prove a stronger result without a connectivity assumption for graphs of high minimum degree, which shows Thomassen's conjecture holds in that case. This result is within a constant factor of best possible. In the process of proving this, we prove a more general result showing that large minimum degree forces a large difference between the order of the largest cycle and the order of the largest chordless cycle.

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Every longest circuit of a 3‐connected, K_3,3‐minor free graph has a chord

Cited by 5 publications

References 7 publications

Geography of Technology Transfer in China

Geography of Technology Transfer in China

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

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Every longest circuit of a 3‐connected, K3,3‐minor free graph has a chord

Cited by 5 publications

References 7 publications

Geography of Technology Transfer in China

Geography of Technology Transfer in China

Removable Edges and Chords of Longest Cycles in 3-Connected Graphs

A Cycle of Maximum Order in a Graph of High Minimum Degree has a Chord

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Every longest circuit of a 3‐connected, K_3,3‐minor free graph has a chord