森林的顶点识别（Vertex identification to a forest）
Vertex identification to a forest
AI总结：
本文研究森林识别问题，证明其为NP-完全，以识别集大小k为参数时存在2k+1大小的核，还分析了相关子式闭类的障碍结构及大图的子式特征。
摘要：
设$\u27e8\frak{H}\u27e9$为一个图类，$k\in\mathbb{N}$。若存在大小至多为$k$的集合$X\subseteq V(G)$的一个划分$\u27e8\frak{P}\u27e9$，使得将$\u27e8\frak{P}\u27e9$中的每个部分识别为单个顶点后，得到的图属于$\u27e8\frak{H}\u27e9$，则称图$G$可实现到$\u27e8\frak{H}\u27e9$的$k$-识别。图参数${\sf id}_{\mathcal{H}}$的定义为：${\sf id}_{\mathcal{H}}(G)$是使得$G$可实现到$\u27e8\frak{H}\u27e9$的$k$-识别的最小$k$，而“到$\u27e8\frak{H}\u27e9$的识别（Identification to $\u27e8\frak{H}\u27e9$）”问题是指给定图$G$和$k\in\mathbb{N}$，判断是否${\sf id}_{\mathcal{H}}(G)\le k$。若将$\u27e8\frak{H}\u27e9$设为无环图类$\u27e8\frak{F}\u27e9$，则得到“森林识别（Identification to Forest）”问题，本文证明该问题是{\uf04e\u0195}-完全的。我们证明，当以识别集的大小$k$为参数时，该问题存在大小为$2k+1$的核。在核的构造中，我们揭示了森林识别问题与顶点覆盖（Vertex Cover）问题的密切关联。我们还研究了到$\u27e8\frak{H}\u27e9$的识别问题的是实例的组合性质，即类$\u27e8\frak{H}\u27e9^{(k)}:=\{G\mid {\sf id}_{\mathcal{H}}(G)\le k\}$，并证明当$\u27e8\frak{H}\u27e9$是子式闭的时，对每个$k$，$\u27e8\frak{H}\u27e9^{(k)}$都是子式闭的。我们证明$\u27e8\frak{F}\u27e9^{(k)}$的子式障碍的大小至多为$2k+4$。我们还证明，每个满足${\sf id}_{\mathcal{F}}(G)$足够大的图$G$，都包含$k$个顶点的环、$k$个不相交的三角形，或者“$k$-雏菊（$k$-marguerite）”图作为子式——$k$-雏菊图是指将$k$个不相交的三角形各取一个顶点识别为同一个顶点所得到的图。
英文摘要：
Let $\mathcal{H}$ be a graph class and $k\in\mathbb{N}$. We say a graph $G$ admits a \emph{$k$-identification to $\mathcal{H}$} if there is a partition $\mathcal{P}$ of some set $X\subseteq V(G)$ of size at most $k$ such that after identifying each part in $\mathcal{P}$ to a single vertex, the resulting graph belongs to $\mathcal{H}$. The graph parameter ${\sf id}_{\mathcal{H}}$ is defined so that ${\sf id}_{\mathcal{H}}(G)$ is the minimum $k$ such that $G$ admits a $k$-identification to $\mathcal{H}$, and the problem of \textsc{Identification to $\mathcal{H}$} asks, given a graph $G$ and $k\in\mathbb{N}$, whether ${\sf id}_{\mathcal{H}}(G)\le k$. If we set $\mathcal{H}$ to be the class $\mathcal{F}$ of acyclic graphs, we generate the problem \textsc{Identification to Forest}, which we show to be {\sf NP}-complete. We prove that, when parameterized by the size $k$ of the identification set, it admits a kernel of size $2k+1$. For our kernel we reveal a close relation of \textsc{Identification to Forest} with the \textsc{Vertex Cover} problem. We also study the combinatorics of the \textsf{yes}-instances of \textsc{Identification to $\mathcal{H}$}, i.e., the class $\mathcal{H}^{(k)}:=\{G\mid {\sf id}_{\mathcal{H}}(G)\le k\}$, {which we show to be minor-closed for every $k$} when $\mathcal{H}$ is minor-closed. We prove that the minor-obstructions of $\mathcal{F}^{(k)}$ are of size at most $2k+4$. We also prove that every graph $G$ such that ${\sf id}_{\mathcal{F}}(G)$ is sufficiently big contains as a minor either a cycle on $k$ vertices, or $k$ disjoint triangles, or the \emph{$k$-marguerite} graph, that is the graph obtained by $k$ disjoint triangles by identifying one vertex of each of them into the same vertex.
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