A complexity dichotomy for the dominating set problem

TL;DR

This paper establishes a complete classification of the computational complexity of the dominating set problem across all hereditary graph classes characterized by forbidden induced subgraphs with up to five vertices.

Contribution

It provides a full complexity dichotomy for the dominating set problem in these graph classes, resolving an open problem in graph theory and computational complexity.

Findings

Identifies which hereditary classes allow polynomial-time solutions.

Proves NP-completeness for classes where the problem remains hard.

Completes the complexity landscape for small forbidden induced subgraphs.

Abstract

We completely determine the complexity status of the dominating set problem for hereditary graph classes defined by forbidden induced subgraphs with at most five vertices.