We introduce graph width parameters, called $\alpha$-edge-crossing width and edge-crossing width. These are defined in terms of the number of edges crossing a bag of a tree-cut decomposition. They are motivated by edge-cut width, recently introduced by Brand et al. (WG 2022). We show that edge-crossing width is equivalent to the known parameter tree-partition-width. On the other hand, $\alpha$-edge-crossing width is a new parameter; tree-cut width and $\alpha$-edge-crossing width are incomparable, and they both lie between tree-partition-width and edge-cut width. We provide an algorithm that, for a given $n$-vertex graph $G$ and integers $k$ and $\alpha$, in time $2^{O((\alpha+k)\log (\alpha+k))}n^2$ either outputs a tree-cut decomposition certifying that the $\alpha$-edge-crossing width of $G$ is at most $2\alpha^2+5k$ or confirms that the $\alpha$-edge-crossing width of $G$ is more than $k$. As applications, for every fixed $\alpha$, we obtain FPT algorithms for the List Coloring and Precoloring Extension problems parameterized by $\alpha$-edge-crossing width. They were known to be W[1]-hard parameterized by tree-partition-width, and FPT parameterized by edge-cut width, and we close the complexity gap between these two parameters.

翻译：我们引入了图宽度参数，称为 $α$-边交叉宽度（α-edge-crossing width）和边交叉宽度（edge-crossing width）。这些参数基于树割分解中穿过一个袋（bag）的边的数量定义。其动机源于 Brand 等人（WG 2022）近期提出的边割宽度（edge-cut width）。我们证明了边交叉宽度等价于已知参数树划分宽度（tree-partition-width）。另一方面，$α$-边交叉宽度是一个新参数；树割宽度（tree-cut width）与 $α$-边交叉宽度不可比较，且两者均介于树划分宽度和边割宽度之间。我们提供了一个算法：对于给定的 $n$ 顶点图 $G$ 以及整数 $k$ 和 $\alpha$，在时间 $2^{O((\alpha+k)\log (\alpha+k))}n^2$ 内，要么输出一个树割分解，证明 $G$ 的 $α$-边交叉宽度至多为 $2\alpha^2+5k$，要么确认 $G$ 的 $α$-边交叉宽度大于 $k$。作为应用，对于每个固定的 $\alpha$，我们得到了以 $α$-边交叉宽度为参数化的列表着色（List Coloring）和预着色扩展（Precoloring Extension）问题的 FPT 算法。已知这些问题在以树划分宽度为参数时是 W[1]-难的，而以边割宽度为参数时是 FPT 的，我们的工作填补了这两个参数之间的复杂度空白。