We study the parameterized complexity of the Bounded-Degree Vertex Deletion problem (BDD), where the aim is to find a maximum induced subgraph whose maximum degree is below a given degree bound. Our focus lies on parameters that measure the structural properties of the input instance. We first show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, treedepth, and even the size of a minimum vertex deletion set into graphs of pathwidth and treedepth at most three. We thereby resolve an open question stated in Betzler, Bredereck, Niedermeier and Uhlmann (2012) concerning the complexity of BDD parameterized by the feedback vertex set number. On the positive side, we obtain fixed-parameter algorithms for the problem with respect to the decompositional parameter treecut width and a novel problem-specific parameter called the core fracture number.

中文翻译：

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关于有界顶点删除问题的结构参数化

我们研究了有界度顶点删除问题 (BDD) 的参数化复杂性，其目的是找到最大度数低于给定度数界限的最大诱导子图。我们的重点在于测量输入实例结构属性的参数。我们首先表明，该问题是 W[1]-hard 参数化的，由一系列相当严格的结构参数（例如反馈顶点集数、路径宽度、树深度，甚至最小顶点删除集到路径宽度和treedepth 最多三个。因此，我们解决了 Betzler、Bredereck、Niedermeier 和 Uhlmann (2012) 中提到的关于由反馈顶点集数参数化的 BDD 的复杂性的一个悬而未决的问题。从积极的方面来说，