# Domination above r-independence: does sparseness help?

## TL;DR

This paper explores the complexity of the Dominating Set problem when given a maximal r-independent set, analyzing how the parameter r and graph sparsity influence tractability and kernelization.

## Contribution

It introduces a new parameterization based on a lower-bound witness and studies its impact on problem complexity across different graph classes.

## Key findings

- For r=2, the problem is paraNP-complete even on sparse graphs.
- For r=3, the problem is W[2]-hard in general but FPT in nowhere dense classes.
- For r>=4, the parameterization aligns with the natural parameter, the size of the dominating set.

## Abstract

Inspired by the potential of improving tractability via gap- or above-guarantee parametrisations, we investigate the complexity of Dominating Set when given a suitable lower-bound witness. Concretely, we consider being provided with a maximal r-independent set X (a set in which all vertices have pairwise distance at least r + 1) along the input graph G which, for r >= 2, lower-bounds the minimum size of any dominating set of G. In the spirit of gap-parameters, we consider a parametrisation by the size of the 'residual' set R := V (G) \ N [X]. Our work aims to answer two questions: How does the constant r affect the tractability of the problem and does the restriction to sparse graph classes help here? For the base case r = 2, we find that the problem is paraNP -complete even in apex- and bounded-degree graphs. For r = 3, the problem is W[2]-hard for general graphs but in FPT for nowhere dense classes and it admits a linear kernel for bounded expansion classes. For r >= 4, the parametrisation becomes essentially equivalent to the natural parameter, the size of the dominating set.