Excluding a Ladder

Huynh, Tony; Joret, Gwenaël; Micek, Piotr; Seweryn, Michał T.; Wollan, Paul

doi:10.1007/s00493-021-4592-8

Cited by 4 publications

(2 citation statements)

References 17 publications

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“…Generally, in the past 15 years alone, more than 25 published research papers have explored connections between dimension and graph theoretic properties of cover graphs and order diagrams (we list some of them: outerplanarity [3,5], planarity [18,13,1,10], cut-vertices structure [22], treewidth [17,9], exclusion of some structures [25,15,6,7], or even more complex structural parameters [11,8]). For our considerations, it is worth mentioning a result in [9].…”

Section: Introductionmentioning

confidence: 99%

Forcing the wheel

Jędrzej

Trotter

2023

Preprint

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Over the past 10 years, there has been considerable interest in exploring questions connecting dimension for posets with graph theoretic properties of their cover graphs and order diagrams, especially with the concepts of planarity and treewidth. Joret and Micek conjectured that if P is a poset with a planar cover graph, then the dimension of P is bounded in terms of the number of minimal elements of P and the treewidth of the cover graph of P. We settle this conjecture in the affirmative. To this end, we strengthen a recent breakthrough result by Blake, Micek, and Trotter, who proved that for each poset P admitting a planar cover graph and a unique minimal element we have dim(P) <= 2 se(P) + 2, where dim(P) stands for the dimension of P and se(P) stands for the maximum order of a standard example contained in P. We prove that dim(P) <= 2 wheel(P) + 2, where wheel(P) stands for the maximum order of a wheel contained in P.

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Section: Introductionmentioning

confidence: 99%

Forcing the wheel

Jędrzej

Trotter

2023

Preprint

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“…The classes of forests and outerplanar graphs do have the property described in the question and the class of planar graphs does not. Other examples of graph classes enjoying that property include graphs of pathwidth at most 2 [1,12], graphs of treewidth at most 2 [4,9], and graphs which exclude a fixed 2 × k grid as a minor [3]. It was conjectured in [3] that the answer to the question above are exactly those classes which exclude the cover graph of a poset in the Kelly's construction.…”

Section: Introductionmentioning

confidence: 99%

Posets with $k$-outerplanar cover graphs have bounded dimension

Gorsky,

Seweryn

2021

Preprint

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In 2015, Felsner, Trotter, and Wiechert showed that posets with outerplanar cover graphs have bounded dimension. We generalise this result to posets with k-outerplanar cover graphs. Namely, we show that posets with k-outerplanar cover graph have dimension O(k 3 ). As a consequence, we show that every poset with a planar cover graph and height h has dimension O(h 3 ). This improves the previously best known bound of O(h 6 ) by Kozik, Micek and Trotter.

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