Given a graph $G$, an integer $k\geq 0$, and a non-negative integral function $f:V(G) \rightarrow \mathcal{N}$, the {\sc Vector Domination} problem asks whether a set $S$ of vertices, of cardinality $k$ or less, exists in $G$ so that every vertex $v \in V(G)-S$ has at least $f(v)$ neighbors in $S$. The problem generalizes several domination problems and it has also been shown to generalize Bounded-Degree Vertex Deletion. In this paper, the parameterized version of Vector Domination is studied when the input graph is planar. A linear problem kernel is presented.

翻译：给定图 $G$、整数 $k\geq 0$ 以及非负整值函数 $f:V(G) \rightarrow \mathcal{N}$，向量支配问题询问在图 $G$ 中是否存在一个基数不超过 $k$ 的顶点子集 $S$，使得每个顶点 $v \in V(G)-S$ 在 $S$ 中至少拥有 $f(v)$ 个邻接顶点。该问题推广了多个支配问题，并已被证明可推广有界度数顶点删除问题。本文研究输入图为平面图时向量支配问题的参数化版本，并给出了一个线性问题核。