
Tree pivotminors and linear rankwidth
Treewidth and its linear variant pathwidth play a central role for the...
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Treewidth dichotomy
We prove that the treewidth of graphs in a hereditary class defined by ...
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Meyniel's conjecture on graphs of bounded degree
The game of Cops and Robbers is a well known pursuitevasion game played...
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The Z_2genus of Kuratowski minors
A drawing of a graph on a surface is independently even if every pair of...
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Twinwidth II: small classes
The twinwidth of a graph G is the minimum integer d such that G has a d...
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A Flat Wall Theorem for Matching Minors in Bipartite Graphs
A major step in the graph minors theory of Robertson and Seymour is the ...
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MSO Undecidability for some Hereditary Classes of Unbounded CliqueWidth
Seese's conjecture for finite graphs states that monadic secondorder lo...
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On the treewidth of evenholefree graphs
The class of all evenholefree graphs has unbounded treewidth, as it contains all complete graphs. Recently, a class of (evenhole, K_4)free graphs was constructed, that still has unbounded treewidth [Sintiari and Trotignon, 2019]. The class has unbounded degree and contains arbitrarily large cliqueminors. We ask whether this is necessary. We prove that for every graph G, if G excludes a fixed graph H as a minor, then G either has small treewidth, or G contains a large wall or the line graph of a large wall as induced subgraph. This can be seen as a strengthening of Robertson and Seymour's excluded grid theorem for the case of minorfree graphs. Our theorem implies that every class of evenholefree graphs excluding a fixed graph as a minor has bounded treewidth. In fact, our theorem applies to a more general class: (theta, prism)free graphs. This implies the known result that planar even holefree graph have bounded treewidth [da Silva and Linhares Sales, Discrete Applied Mathematics 2010]. We conjecture that evenholefree graphs of bounded degree have bounded treewidth. If true, this would mean that evenholefreeness is testable in the boundeddegree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the treewidth of the class of (even hole, pyramid)free graphs of degree at most 4.
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