Non-separating induced cycles in graphs

Thomassen, Carsten; Toft, Bjarne

doi:10.1016/s0095-8956(81)80025-1

Cited by 75 publications

(50 citation statements)

References 14 publications

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“…A cycle C of a connected graph H is called removable if the graph H − E(C) is connected, and non-separating if H − V (C) is connected. The following result is due to Thomassen and Toft [26].…”

Section: Closed Trails Of Low Degeneracy and Orderingmentioning

confidence: 87%

Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

et al. 2011

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(2013) 'Exact algorithms for nding longest cycles in claw-free graphs.', Algorithmica., 65 (1). 129 -145. Further information on publisher's website:http://dx.doi.org/10.1007/s00453-011-9576-4Publisher's copyright statement:The original publication is available at www.springerlink.com Additional information: Use policyThe full-text may be used and/or reproduced, and given to third parties in any format or medium, without prior permission or charge, for personal research or study, educational, or not-for-prot purposes provided that:• a full bibliographic reference is made to the original source • a link is made to the metadata record in DRO • the full-text is not changed in any way The full-text must not be sold in any format or medium without the formal permission of the copyright holders.Please consult the full DRO policy for further details. Abstract. The Hamiltonian Cycle problem is the problem of deciding whether an n-vertex graph G has a cycle passing through all vertices of G. This problem is a classic NP-complete problem. Finding an exact algorithm that solves it in O

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Section: Closed Trails Of Low Degeneracy and Orderingmentioning

confidence: 87%

Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

et al. 2011

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“…Hence, by (1) - (4) and by (v) of Lemma (2.4), we may assume from symmetry that H 1 = ∅ = W 3 . Then H 2 ∪ Q 2 ∪ Q 1 and W 2 ∪ Q 2 ∪ Q 3 are cuts in G. Since V (K) ⊆ Q 2 ∪ H 2 , it follows from the choice of S (see (5)) that |H 2 ∪ Q 2 ∪ Q 1 | ≥ |S| + 1. Since W 2 ∪ Q 2 ∪ Q 3 is a cut in G, |W 2 ∪ Q 2 ∪ Q 3 | ≥ k = |S |.…”

Section: By (1) -(4) and Bymentioning

confidence: 99%

“…This result is then used in [5] to prove a conjecture of Lovász. Extending techniques of Egawa [1], Kawarabayashi [3] improved Thomassen's result by showing that for k ≥ 4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge not contained in any triangle, or admits a k-contractible triangle which does not share an edge with any other triangle.…”

Section: Introductionmentioning

confidence: 99%

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Contractible Cliques in k-Connected Graphs

Huang

Jin

et al. 2006

Graphs and Combinatorics

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Kawarabayashi proved that for any integer k ≥ 4, every k-connected graph contains two triangles sharing an edge, or admits a k-contractible edge, or admits a k-contractible triangle. This implies Thomassen's result that every triangle-free k-connected graph contains a k-contractible edge. In this paper, we extend Kawarabayashi's technique and prove a more general result concerning k-contractible cliques.

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“…By the same example, Thomassen [12,13] conjectured that every (a + b + 1)-connected graph can be decomposed into two parts A and B in such a way that A is a-connected and B is b-connected. It was shown by Thomassen himself [10] We only know that f (1, 1) = 3 holds by a result of Thomassen [11]. No much progress has been made for this problem.…”

Section: Introductionmentioning

confidence: 99%

Decomposing edge-colored graphs under color degree constraints

Fujita

Wang

2017

Electronic Notes in Discrete Mathematics

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For an edge-colored graph G, the minimum color degree of G means the minimum number of colors on edges which are adjacent to each vertex of G. We prove that if G is an edge-colored graph with minimum color degree at least 5 then V(G) can be partitioned into two parts such that each part induces a subgraph with minimum color degree at least 2. We show this theorem by proving a much stronger form. Moreover, we point out an important relationship between our theorem and Bermond-Thomassen's conjecture in digraphs.

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Non-separating induced cycles in graphs

Cited by 75 publications

References 14 publications

Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

Exact Algorithms for Finding Longest Cycles in Claw-Free Graphs

Contractible Cliques in k-Connected Graphs

Decomposing edge-colored graphs under color degree constraints

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