Parameterized (Approximate) Defective Coloring
R\'emy Belmonte, Michael Lampis, Valia Mitsou

TL;DR
This paper explores the complexity and approximability of defective coloring in graphs, revealing fixed-parameter tractability for feedback vertex set and establishing ETH-based lower bounds for treewidth and pathwidth.
Contribution
It provides new fixed-parameter algorithms for defective coloring parameterized by feedback vertex set and proves hardness results for other parameters, along with approximation algorithms and bounds.
Findings
W-hardness for treewidth, pathwidth, tree-depth, feedback vertex set when χ_d=2
FPT algorithm for feedback vertex set for any χ_d ≥ 2
ETH-based lower bounds for treewidth and pathwidth
Abstract
In Defective Coloring we are given a graph and two integers and are asked if we can partition into color classes, so that each class induces a graph of maximum degree . We investigate the complexity of this generalization of Coloring with respect to several well-studied graph parameters, and show that the problem is W-hard parameterized by treewidth, pathwidth, tree-depth, or feedback vertex set, if . As expected, this hardness can be extended to larger values of for most of these parameters, with one surprising exception: we show that the problem is FPT parameterized by feedback vertex set for any , and hence 2-coloring is the only hard case for this parameter. In addition to the above, we give an ETH-based lower bound for treewidth and pathwidth, showing that no algorithm can solve the problem in…
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