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arXiv · 2312.09374

A Linear Kernel for Planar Vector Domination

Mahabba El Sahili·Faisal N. Abu-Khzam
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Abstract

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.

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Mahabba El Sahili, Faisal N. Abu-Khzam. 2025-06-18. A Linear Kernel for Planar Vector Domination. https://arxiv.org/abs/2312.09374

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