Vertex deletion to hereditary graph class is well-studied in parameterized complexity. Vertex deletion to the scattered graph classes has gained attention in recent years. In this paper, we consider (Proper-Interval, Tree)-Vertex Deletion, the input to which is an undirected graph $G = (V, E)$ and an integer $k$. The goal is to pick a set $X \subseteq V(G)$ of at most $k$ vertices such that $G - X$ is a simple graph and every connected component of $G - X$ is a proper interval graph or a tree. When parameterized by the solution size $k$, (Proper-Interval, Tree)-Vertex Deletion has been proved to be fixed-parameter tractable by Jacob et al. [JCSS-2023, FCT-2021]. In this paper, we consider this problem from the perspective of polynomial kernelization. We provide a first nontrivial polynomial kernel for (Proper-Interval, Tree)-Vertex Deletion, with $O(k^{33})$ vertices.

翻译：顶点删除至遗传图类在参数化复杂性中被广泛研究。近年来，顶点删除至稀疏图类引起了关注。本文研究（真区间图，树）-顶点删除问题，其输入为无向图$G = (V, E)$和整数$k$。目标是选取至多$k$个顶点构成集合$X \subseteq V(G)$，使得$G - X$是简单图，且$G - X$的每个连通分量要么是真区间图，要么是树。当以解的大小$k$为参数时，（真区间图，树）-顶点删除已被Jacob等人[JCSS-2023, FCT-2021]证明为固定参数可解。本文从多项式核化角度考虑该问题，首次对（真区间图，树）-顶点删除问题给出了非平凡的多项式核，其顶点数为$O(k^{33})$。