AI 中文总结
研究H-填充问题，通过归约和两步证明，为细分星图等特定模式图得到不同规模的核，同时证明S_0,d的下界，表明删除模式图单个顶点可能使核化更难，揭示H-填充可压缩性在诱导子图下非单调。
AI 中文摘要
H-填充问题是询问图G是否包含k个固定模式图H的顶点不相交副本。通过标准归约到d-集填充，可得到具有O(k^|V(H)|-1)个顶点和O(k^|V(H)|)条边的通用核。本文重新探讨针对特定模式H改进这些界限的问题。主要结果涉及细分星图。对于P_5 = S_0,2、S_1,2和每个S_d1,1，得到具有O(k^2)个顶点和O(k^3)条边的核；对于每个固定的d_1≥1的S_d1,d2，得到具有O(k^4)个顶点和O(k^6)条边的核；对于爪形图，得到具有O(k^2)个顶点和O(k^4)条边的核。证明分两步，首先归约到图的大部分是独立的或有小顶点覆盖的实例，然后通过为小部分的每个子集保留有限数量的见证顶点来减少独立部分。负面结果是证明了S_0,d的下界，对于每个d≥3和每个ε>0，除非NP⊆coNP/poly，S_0,d-填充不允许大小为O(k^d-ε)的压缩。这表明从模式中删除单个顶点可能使核化更难，即H-填充的可压缩性在取诱导子图时不是单调的。
英文摘要
\textsc{$H$-Packing} asks whether a graph $G$ contains $k$ vertex-disjoint copies of a fixed pattern graph $H$. Via the standard reduction to \textsc{$d$-Set Packing}, one obtains generic kernels with $O(k^{|V(H)|-1})$ vertices and $O(k^{|V(H)|})$ edges. We revisit the question of beating these bounds for specific patterns $H$. Our main results concern subdivided stars. Let $S_{d_1,d_2}$ denote the subdivided star with $d_1$ branches of length $1$ and $d_2$ branches of length $2$. We obtain kernels with $O(k^2)$ vertices and $O(k^3)$ edges for $P_5=S_{0,2}$, for $S_{1,2}$, and for every $S_{d_1,1}$, kernels with $O(k^4)$ vertices and $O(k^6)$ edges for every fixed $S_{d_1,d_2}$ with $d_1\ge 1$, and a kernel with $O(k^2)$ vertices and $O(k^4)$ edges for the paw. Our proofs proceed in two steps. First, we reduce to instances in which all but a small part of the graph is independent, or in which the graph has a small vertex cover. Second, we reduce the independent side by keeping only a bounded number of witness vertices for each subset of the small part. On the negative side, we prove a lower bound for the line $S_{0,d}$. For every $d\ge 3$ and every $\varepsilon>0$, \textsc{$S_{0,d}$-Packing} does not admit a compression of size $O(k^{d-\varepsilon})$ unless $\NP\subseteq \coNP/\poly$. Thus, deleting a single vertex from the pattern may, surprisingly, make kernelization provably harder, showing that compressibility of \textsc{$H$-Packing} is not monotone under taking induced subgraphs.
CommentsESA 2026
