无H边修改问题的不可压缩性：走向二分法
Incompressibility of H-free edge modification problems: Towards a dichotomy
摘要：
给定一个图$G$和一个整数$k$，无$H$边编辑问题是要找出在$G$中是否存在最多$k$对顶点，使得改变$G$中的顶点对的邻接性导致一个图没有任何$H$的诱导副本. $H$ -free Edge Editing的多项式核的存在性在参数化复杂性文献中受到了极大的关注。已知对于至多4个顶点的某些图$H$存在非平凡多项式核，但是从5个顶点开始，多项式核只有当$H$是完全的或空的时才是已知的。这表明了这样的猜想，即没有其他具有至少5个顶点的$H$是无$H$的边缘编辑允许多项式核。为了实现这个目标，我们得到了一组9个5顶点图$\mathcal{H}$，使得如果对于每个$H\in\mathcal{H}$，$H$ -free Edge Editing是不可压缩的，并且复杂性假设$NP \not\subseteq coNP/poly$成立，则对于每个具有至少5个顶点的图$H$，$H$ -free Edge Editing是不可压缩的，该图既不完整也不为空。也就是说，证明这九个图的不可压缩性将为每个具有至少5个顶点的$H$给出无$H$边编辑的核化复杂度的完整分类。对于无$H$的边删除，我们也得到了类似的结果.这里的情况更加复杂，因为存在另一个无限族的图$H$，其中问题是微不足道的（只有一条边的图）。我们得到了一个更大的集合$\mathcal{H}$的十九个图的不可压缩性将给出一个完整的分类的核化复杂度的$H$ -free边删除的每一个图$H$至少有5个顶点。类似的结果也遵循$H$ -免费的边缘完成问题，通过简单的补充。
英文摘要：
Given a graph $G$ and an integer $k$, the $H$-free Edge Editing problem is to find whether there exists at most $k$ pairs of vertices in $G$ such that changing the adjacency of the pairs in $G$ results in a graph without any induced copy of $H$. The existence of polynomial kernels for $H$-free Edge Editing received significant attention in the parameterized complexity literature. Nontrivial polynomial kernels are known to exist for some graphs $H$ with at most 4 vertices, but starting from 5 vertices, polynomial kernels are known only if $H$ is either complete or empty. This suggests the conjecture that there is no other $H$ with at least 5 vertices were $H$-free Edge Editing admits a polynomial kernel. Towards this goal, we obtain a set $\mathcal{H}$ of nine 5-vertex graphs such that if for every $H\in\mathcal{H}$, $H$-free Edge Editing is incompressible and the complexity assumption $NP \not\subseteq coNP/poly$ holds, then $H$-free Edge Editing is incompressible for every graph $H$ with at least five vertices that is neither complete nor empty. That is, proving incompressibility for these nine graphs would give a complete classification of the kernelization complexity of $H$-free Edge Editing for every $H$ with at least 5 vertices. We obtain similar result also for $H$-free Edge Deletion. Here the picture is more complicated due to the existence of another infinite family of graphs $H$ where the problem is trivial (graphs with exactly one edge). We obtain a larger set $\mathcal{H}$ of nineteen graphs whose incompressibility would give a complete classification of the kernelization complexity of $H$-free Edge Deletion for every graph $H$ with at least 5 vertices. Analogous results follow also for the $H$-free Edge Completion problem by simple complementation.
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