# Complexity of fall coloring for restricted graph classes

## TL;DR

This paper investigates the computational complexity of fall coloring in restricted graph classes, establishing NP-completeness results for various graph partitioning problems and providing algorithms for certain cases.

## Contribution

It extends known NP-completeness results for graph partitioning into independent dominating sets and introduces algorithms for specific graph classes.

## Key findings

- NP-complete to decide partitioning into three independent dominating sets for bipartite planar graphs
- Partitioning into k independent dominating sets is NP-complete for all k ≥ 3 in bipartite graphs
- Deciding two disjoint independent dominating sets is NP-complete even for triangle-free planar graphs

## Abstract

We strengthen a result by Laskar and Lyle (Discrete Appl. Math. (2009), 330-338) by proving that it is NP-complete to decide whether a bipartite planar graph can be partitioned into three independent dominating sets. In contrast, we show that this is always possible for every maximal outerplanar graph with at least three vertices. Moreover, we extend their previous result by proving that deciding whether a bipartite graph can be partitioned into $k$ independent dominating sets is NP-complete for every $k \geq 3$. We also strengthen a result by Henning et al. (Discrete Math. (2009), 6451-6458) by showing that it is NP-complete to determine if a graph has two disjoint independent dominating sets, even when the problem is restricted to triangle-free planar graphs. Finally, for every $k \geq 3$, we show that there is some constant $t$ depending only on $k$ such that deciding whether a $k$-regular graph can be partitioned into $t$ independent dominating sets is NP-complete. We conclude by deriving moderately exponential-time algorithms for the problem.