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On the tree-width of even-hole-free graphs
European Journal of Combinatorics ( IF 1.0 ) Pub Date : 2021-07-12 , DOI: 10.1016/j.ejc.2021.103394
Pierre Aboulker 1 , Isolde Adler 2 , Eun Jung Kim 3 , Ni Luh Dewi Sintiari 4 , Nicolas Trotignon 4
Affiliation  

The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, K4)-free graphs was constructed, that still has unbounded tree-width (Sintiari and Trotignon, 2019). The class has unbounded degree and contains arbitrarily large clique-minors. 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 tree-width, 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 minor-free graphs. Our theorem implies that every class of even-hole-free graphs excluding a fixed graph as a minor has bounded tree-width. In fact, our theorem applies to a more general class: (theta, prism)-free graphs. This implies the known result that planar even hole-free graphs have bounded tree-width (Silva et al., 2010).

We conjecture that even-hole-free graphs of bounded degree have bounded tree-width. If true, this would mean that even-hole-freeness is testable in the bounded-degree graph model of property testing. We prove the conjecture for subcubic graphs and we give a bound on the tree-width of the class of (even hole, pyramid)-free graphs of degree at most 4.



中文翻译:

关于无偶数孔图的树宽

所有无偶数孔图的类都具有无限的树宽,因为它包含所有完整的图。最近,一类(偶数孔,4)-free 图被构建,它仍然具有无限的树宽(Sintiari 和 Trotignon,2019)。该类具有无限度包含任意大的小集团。我们问这是否有必要。

我们证明对于每个图 G, 如果 G 排除固定图 H 作为未成年人,那么 G 要么树宽小,要么 G包含大墙或大墙的线图作为诱导子图。这可以看作是对 Robertson 和 Seymour 的排除网格定理的加强,适用于无次要图的情况。我们的定理意味着,除了作为次要的固定图之外,每类无偶数孔图都具有有界树宽。事实上,我们的定理适用于更一般的类:(theta,prism)-free 图。这意味着已知的结果是平面偶数无孔图具有有界树宽(Silva 等,2010)。

我们推测有界度的偶数无孔图具有有界树宽。如果为真,这将意味着偶数孔自由度在属性测试的有界度图模型中是可测试的。我们证明了亚立方图的猜想,并给出了度数最多为 4 的无(偶数孔、金字塔)图的树宽的界限。

更新日期:2021-07-12
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