$w$-Dominating Set Problem on Graphs of Bounded Treewidth

TL;DR

This paper presents polynomial time algorithms for the $w$-Dominating Set and $L$-Max $w$-Dominating Set problems on graphs with bounded treewidth, advancing efficient solutions for these graph problems.

Contribution

It introduces the first polynomial time algorithms for these problems specifically on graphs of bounded treewidth, a significant theoretical advancement.

Findings

Polynomial algorithms for $w$-Dominating Set.

Polynomial algorithms for $L$-Max $w$-Dominating Set.

Applicable to graphs with bounded treewidth.

Abstract

Let $G=(V,E)$ be a graph. Let $w$ be a positive integer. A $w$-dominating set is a vertex subset $S$ such that for all $v\in V$, either $v\in S$ or it has at least $w$ neighbors in $S$. The $w$-Dominating Set problem is to find the minimum $w$-dominating set. The $L$-Max $w$-Dominating Set problem is to find the vertex subset $S$ of cardinality at most $L$ that maximizes $|S|+|\{v\in V\setminus S~|~|N(v)\cap S|\geq w\}|$, where $N(v)=\{u|uv\in E\}$. In this paper, we give polynomial time algorithms to $w$-Dominating Set problem and $L$-Max $w$-Dominating Set problem on graphs of bounded treewidth.