On Complexities of Minus Domination

TL;DR

This paper investigates the computational complexity of the minus-domination problem in various graph classes, establishing fixed-parameter tractability for some and NP-completeness for others, highlighting the problem's nuanced difficulty landscape.

Contribution

The paper characterizes the complexity of minus-domination across different graph classes, providing new fixed-parameter algorithms and NP-completeness results.

Findings

Fixed-parameter tractability for d-degenerate graphs

Polynomial algorithms for graphs of bounded rankwidth and strongly chordal graphs

NP-completeness for splitgraphs

Abstract

A function f: V \rightarrow \{-1,0,1\} is a minus-domination function of a graph G=(V,E) if the values over the vertices in each closed neighborhood sum to a positive number. The weight of f is the sum of f(x) over all vertices x \in V. The minus-domination number \gamma^{-}(G) is the minimum weight over all minus-domination functions. The size of a minus domination is the number of vertices that are assigned 1. In this paper we show that the minus-domination problem is fixed-parameter tractable for d-degenerate graphs when parameterized by the size of the minus-dominating set and by d. The minus-domination problem is polynomial for graphs of bounded rankwidth and for strongly chordal graphs. It is NP-complete for splitgraphs. Unless P=NP there is no fixed-parameter algorithm for minus-domination. 79,1 5%