A Linear Kernel for Planar Vector Domination

Given a graph $G$, an integer $k\geq 0$, and a non-negative integral function $f:V(G) \rightarrow \mathcal{N}$, the Vector Domination problem asks whether a set $S$ of vertices, of cardinality $k$ or less, exists in $G$ so that every vertex $v \in V(G)\setminus S$ 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 (BDVD). In this paper, the parameterized version of Vector Domination is studied when the input graph is planar. A linear problem kernel is presented. A direct consequence is a kernel bound for BDVD that is linear in the parameter $k$ only. Previously known bounds are functions of both the target degree and the input parameter.