Vertex identification to a forest

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.
Comment: 18 pages, 5 figures