SIAM Journal on Discrete Mathematics, Volume 35, Issue 1, Page 105-151, January 2021.
Suppose ${\mathcal{F}}$ is a finite family of graphs. We consider the following meta-problem, called $\mathcal{F}$-Immersion Deletion: given a graph $G$ and integer $k$, decide whether the deletion of at most $k$ edges of $G$ can result in a graph that does not contain any graph from $\mathcal{F}$ as an immersion. This problem is a close relative of the $\mathcal{F}$-Minor Deletion problem studied by Fomin et al. [Proceedings of FOCS, IEEE, 2012, pp. 470--479], where one deletes vertices in order to remove all minor models of graphs from $\mathcal{F}$. We prove that whenever all graphs from $\mathcal{F}$ are connected and at least one graph of $\mathcal{F}$ is planar and subcubic, then the $\mathcal{F}$-Immersion Deletion problem admits a constant-factor approximation algorithm running in time $\mathcal{O}(m^3 \cdot n^3 \cdot \log m)$, a linear kernel that can be computed in time $\mathcal{O}(m^4 \cdot n^3 \cdot \log m)$, and a $\mathcal{O}(2^{\mathcal{O}(k)} + m^4 \cdot n^3 \cdot \log m)$-time fixed-parameter algorithm, where $n,m$ count the vertices and edges of the input graph. These results mirror the findings of Fomin et al., who obtained a similar set of algorithmic results for $\mathcal{F}$-Minor Deletion, under the assumption that at least one graph from $\mathcal{F}$ is planar. An important difference is that we are able to obtain a linear kernel for $\mathcal{F}$-Immersion Deletion, while the exponent of the kernel of Fomin et al. for $\mathcal{F}$-Minor Deletion depends heavily on the family $\mathcal{F}$. In fact, this dependence is unavoidable under plausible complexity assumptions, as proven by Giannopoulou et al. [ACM Trans. Algorithms, 13 (2017), p. 35]. This reveals that the kernelization complexity of $\mathcal{F}$-Immersion Deletion is quite different from that of $\mathcal{F}$-Minor Deletion.

中文翻译：

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浸入式封闭图类边缘删除问题的线性核

SIAM 离散数学杂志，第 35 卷，第 1 期，第 105-151 页，2021 年 1 月。
假设 ${\mathcal{F}}$ 是一个有限的图族。我们考虑以下元问题，称为 $\mathcal{F}$-Immersion Deletion：给定一个图 $G$ 和整数 $k$，决定是否删除至多 $k$$G$ 的边可以导致不包含任何来自 $\mathcal{F}$ 的图作为浸入式。这个问题是 Fomin 等人研究的 $\mathcal{F}$-Minor Deletion 问题的近亲。[FOCS 论文集，IEEE，2012，第 470--479 页]，其中删除顶点以从 $\mathcal{F}$ 中删除所有次要的图模型。我们证明只要 $\mathcal{F}$ 的所有图都是连通的，并且 $\mathcal{F}$ 的至少一个图是平面和亚立方的，那么 $\mathcal{F}$-Immersion Deletion 问题承认一个常数-因子逼近算法运行时间 $\mathcal{O}(m^3 \cdot n^3 \cdot \log m)$, 一个可以及时计算的线性核 $\mathcal{O}(m^4 \cdot n^3 \cdot \log m)$，和一个 $\mathcal{O}(2^{\mathcal{O}( k)} + m^4 \cdot n^3 \cdot \log m)$-time 固定参数算法，其中 $n,m$ 计算输入图的顶点和边。这些结果反映了 Fomin 等人的发现，他们获得了一组类似的 $\mathcal{F}$-Minor Deletion 算法结果，假设 $\mathcal{F}$ 中至少有一个图是平面的。一个重要的区别是，我们能够获得 $\mathcal{F}$-Immersion Deletion 的线性核，而 Fomin 等人的核指数。对于 $\mathcal{F}$-Minor Deletion 严重依赖于家族 $\mathcal{F}$。事实上，正如 Giannopoulou 等人所证明的那样，在合理的复杂性假设下，这种依赖性是不可避免的。[ACM 翻译。算法, 13 (2017), p. 35]。