Playing with parameters: structural parameterization in graphs

TL;DR

This paper explores the parameterized complexity of seven fundamental graph problems when parameterized by various natural parameters, extending the understanding of how different solution sizes influence computational complexity.

Contribution

It systematically analyzes the complexity of multiple graph problems under different parameters, contributing to the field of parameterized ecology.

Findings

Identifies complexity classifications for each problem-parameter pair.

Provides a comprehensive map of fixed-parameter tractability and hardness.

Highlights interactions between different graph parameters.

Abstract

When considering a graph problem from a parameterized point of view, the parameter chosen is often the size of an optimal solution of this problem (the "standard" parameter). A natural subject for investigation is what happens when we parameterize such a problem by various other parameters, some of which may be the values of optimal solutions to different problems. Such research is known as parameterized ecology. In this paper, we investigate seven natural vertex problems, along with their respective parameters: the size of a maximum independent set, the size of a minimum vertex cover, the size of a maximum clique, the chromatic number, the size of a minimum dominating set, the size of a minimum independent dominating set and the size of a minimum feedback vertex set. We study the parameterized complexity of each of these problems with respect to the standard parameter of the others.