AI 中文总结
该研究针对路径宽-η删除问题，打破树宽与路径宽表现相似的预期，证明其在三种参数化情形下存在统一多项式核，并推测以解大小k为参数时也存在统一核。
AI 中文摘要
对于常数η≥0，路径宽-η删除问题是指：给定图G和整数k，是否存在顶点集S⊆V(G)，其大小至多为k，使得G−S的路径宽至多为η。树宽-η删除问题和树深-η删除问题分别针对参数树宽和树深以类似方式定义。Fomin等人[FOCS,2012]的里程碑结果表明，对于任意常数η，这三个问题都存在顶点数为O(k^c(η))的核，其中c(η是依赖于η的常数。Giannopoulou等人[ACM TALG,2017]证明，在某种意义上，该结果对于树宽-η删除问题是最优的：当η≥2时，即使以输入图的顶点覆盖M的大小为参数，也不存在大小为O(|M|^((η+1)/2−ε))的核，其中ε>0。与此结果形成对比的是，他们证明树深-η删除问题存在统一多项式核，即大小为O(k^c)的核，其中c是与η无关的常数。相比之下，路径宽-η删除问题是否存在统一多项式核的问题在文献中被忽视了。由于树宽和路径宽往往表现相似，自然会预期当以顶点覆盖的大小为参数时不存在统一核。令人惊讶的是，我们证明情况并非如此。更具体地说，我们证明当以以下参数化时，路径宽-η删除问题存在统一多项式核：(1)解的大小k加上使得G−M具有有界树深的集合M的大小；(2)到路径宽-1图类的（顶点删除）距离；(3)到树深至多为η+1的图类的距离。这使我们推测，路径宽-η删除问题以解的大小k为参数时存在统一核。
英文摘要
For a constant $η\geq 0$, Pathwidth-$η$ Deletion is the problem of deciding whether, for a given graph $G$ and integer $k$, there is a set $S \subseteq V(G)$ of size at most $k$ such that the pathwidth of $G - S$ is at most $η$. The problems Treewidth-$η$ Deletion and Treedepth-$η$ Deletion are defined similarly for the parameters treewidth and treedepth, respectively. A landmark result of Fomin et al. [FOCS, 2012] shows that, for any constant $η$, all three problems admit a kernel on $O(k^{c(η)})$ vertices, where $c(η)$ is a constant depending on $η$. Giannopoulou et al. [ACM TALG, 2017] show that, in some sense, this result is optimal for Treewidth-$η$ Deletion: for $η\geq 2$ and even when parameterizing by the size of a vertex cover $M$ of the input graph, there is no kernel of size $O(|M|^{\frac{η+1}{2}-\varepsilon})$, for any $\varepsilon > 0$. Contrasting this result, they prove that Treedepth-$η$ Deletion admits a uniform polynomial kernel, that is, a kernel of size $O(k^c)$ for a constant $c$ that is independent of $η$. In comparison, the question whether Pathwidth-$η$ Deletion admits a uniform polynomial kernel has been neglected in the literature. As treewidth and pathwidth tend to behave similarly, it is natural to expect that no uniform kernel exists when parameterizing by the size of a vertex cover. Surprisingly, we show this not to be the case. More concretely, we prove the existence of a uniform polynomial kernel for Pathwidth-$η$ Deletion when parameterizing by (1) the solution size $k$ plus the size of a set $M$ such that $G - M$ has bounded treedepth; (2) the (vertex-deletion) distance to pathwidth-$1$ graphs; (3) the distance to the class of graphs with treedepth at most $η+ 1$. This leads us to conjecture that Pathwidth-$η$ Deletion admits a uniform kernel when parameterizing by the solution size $k$.
