A generalization of Dirac's theorem: Subdivisions of wheels

Turner, Galen E.

doi:10.1016/j.disc.2005.05.003

Cited by 5 publications

(8 citation statements)

References 2 publications

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“…Note that the result stated in [17] is slightly weaker than Theorem 1.1, but the proof given by Turner in [17] exactly proves the version given here. We still include our proof that wheel-free graphs have vertices of degree at most 3 in Section 4.…”

Section: Introductionsupporting

confidence: 62%

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Wheel-free planar graphs

Aboulker

Chudnovsky

Seymour

et al. 2015

European Journal of Combinatorics

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Section: Introductionsupporting

confidence: 62%

“…Thanks to Louis Esperet and Matěj Stehlík for useful discussions and for pointing out to us a graph with chromatic number 4, no triangle and no wheel as an induced subgraph. Thanks to Frédéric Maffray for pointing out to us the paper of Turner [17].…”

Section: Acknowledgementmentioning

confidence: 99%

Wheel-free planar graphs

Aboulker

Chudnovsky

Seymour

et al. 2015

European Journal of Combinatorics

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“…Nebeský [18] proved that every critically 2-connected graph contains a node of degree 2, but critically 2connected graphs are not 3-colorable in general, since they may contain any subgraph of arbitrarily large chromatic number. Note that studying longest paths to obtain nodes of small degree in graphs where "propellerlike" structures are excluded can give much stronger results, see [28].…”

Section: Lemma 210 a Graph G Does Not Contain A Propeller If And Only...mentioning

confidence: 99%

Graphs That Do Not Contain a Cycle with a Node That Has at Least Two Neighbors on It

Aboulker

Radovanović²,

Trotignon

et al. 2012

SIAM J. Discrete Math.

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We recall several known results about minimally 2-connected graphs, and show that they all follow from a decomposition theorem. Starting from an analogy with critically 2-connected graphs, we give structural characterizations of the classes of graphs that do not contain as a subgraph and as an induced subgraph, a cycle with a node that has at least two neighbors on the cycle. From these characterizations we get polynomial time recognition algorithms for these classes, as well as polynomial time algorithms for vertex-coloring and edge-coloring.

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“…• Turner [35] shows that if every vertex of a graph G has degree at least k, then G contains a W k -subdivision, and that every graph with a chromatic number greater than n contains a W n -subdivision. • A fixed vertex version of the subgraph homeomorphism problem for directed graphs, where the input specifies which vertex of G corresponds to each vertex of H , has been completely classified by Fortune, Hopcroft and Wyllie [12].…”

Section: Related Workmentioning

confidence: 99%

Structure and Recognition of Graphs with No 6-wheel Subdivision

Robinson

Farr

2008

Algorithmica

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The subgraph homeomorphism problem has been shown by Robertson and Seymour to be polynomial-time solvable for any fixed pattern graph H . The result, however, is not practical, involving constants that are worse than exponential in |H |. Practical algorithms have only been developed for a few specific pattern graphs, the most recent of these being the wheels with four and five spokes. This paper looks at the subgraph homeomorphism problem where the pattern graph is a wheel with six spokes. The main result is a theorem characterizing graphs that do not contain subdivisions of W 6 . We give an efficient algorithm for solving the subgraph homeomorphism problem for W 6 . We also give a strengthening of the previous W 5 result.

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A generalization of Dirac's theorem: Subdivisions of wheels

Cited by 5 publications

References 2 publications

Wheel-free planar graphs

Wheel-free planar graphs

Graphs That Do Not Contain a Cycle with a Node That Has at Least Two Neighbors on It

Structure and Recognition of Graphs with No 6-wheel Subdivision

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