(Total) Vector Domination for Graphs with Bounded Branchwidth

TL;DR

This paper proves that vector domination problems can be solved efficiently in graphs with bounded branchwidth, including planar graphs, by leveraging their structural properties.

Contribution

The paper introduces a polynomial-time algorithm for (total) vector domination in graphs with bounded branchwidth, extending to graphs with bounded treewidth and planar graphs.

Findings

Polynomial-time solution for bounded branchwidth graphs

Extension to bounded treewidth graphs

Subexponential fixed-parameter tractability for planar graphs

Abstract

Given a graph $G=(V,E)$ of order $n$ and an $n$-dimensional non-negative vector $d=(d(1),d(2),\ldots,d(n))$, called demand vector, the vector domination (resp., total vector domination) is the problem of finding a minimum $S\subseteq V$ such that every vertex $v$ in $V\setminus S$ (resp., in $V$) has at least $d(v)$ neighbors in $S$. The (total) vector domination is a generalization of many dominating set type problems, e.g., the dominating set problem, the $k$-tuple dominating set problem (this $k$ is different from the solution size), and so on, and its approximability and inapproximability have been studied under this general framework. In this paper, we show that a (total) vector domination of graphs with bounded branchwidth can be solved in polynomial time. This implies that the problem is polynomially solvable also for graphs with bounded treewidth. Consequently, the (total) vector domination problem for a planar graph is subexponential fixed-parameter tractable with respectto $k$, where $k$ is the size of solution.