Tree-cut width is a graph parameter introduced by Wollan that is an analogue of treewidth for the immersion order on graphs in the following sense: the tree-cut width of a graph is functionally equivalent to the largest size of a wall that can be found in it as an immersion. In this work we propose a variant of the definition of tree-cut width that is functionally equivalent to the original one, but for which we can state and prove a tight duality theorem relating it to naturally defined dual objects: appropriately defined brambles and tangles. Using this result we also propose a game characterization of tree-cut width.

中文翻译：

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对树切割分解对偶的对象

树形切割宽度是Wollan引入的图形参数，在以下意义上类似于树形的浸入顺序树形宽度：图的树形切割宽度在功能上等同于可以找到的墙的最大尺寸作为沉浸在其中。在这项工作中，我们提出了树形切割宽度定义的一种变体，该变体在功能上与原始的相同，但是我们可以陈述并证明紧密的对偶定理，该定理与自然定义的对偶对象相关：适当定义的荆棘和缠结。使用此结果，我们还提出了对树木砍伐宽度进行游戏表征的方法。