Upper Domination: towards a dichotomy through boundary properties

TL;DR

This paper investigates the computational complexity of finding upper dominating sets in graphs, establishing a dichotomy for classes defined by a single forbidden induced subgraph and identifying the first boundary class.

Contribution

It introduces the first boundary class for the upper dominating set problem and proves a complexity dichotomy for graph classes with one forbidden induced subgraph.

Findings

Identified the first boundary class for the problem

Proved NP-hardness or polynomial-time solvability dichotomy

Established complexity results for classes defined by a single forbidden subgraph

Abstract

An upper dominating set in a graph is a minimal (with respect to set inclusion) dominating set of maximum cardinality. The problem of finding an upper dominating set is generally NP-hard. We study the complexity of this problem in classes of graphs defined by finitely many forbidden induced subgraphs and conjecture that the problem admits a dichotomy in this family, i.e. it is either NP-hard or polynomial-time solvable for each class in the family. A helpful tool to study the complexity of an algorithmic problem on finitely defined classes of graphs is the notion of boundary classes. However, none of such classes has been identified so far for the upper dominating set problem. In the present paper, we discover the first boundary class for this problem and prove the dichotomy for classes defined by a single forbidden induced subgraph.