On the structural parameterized complexity of defective coloring - Bhyravarapu, Sriram; Kumar, Pankaj; Saurabh, Saket (Journal of computer and system sciences, 2026)
In this paper, we consider the problem Defective Coloring. Given a graph G and two positive integers k and Δ⁎, the objective is to determine whether it is possible to obtain a coloring (not necessarily proper) of the vertices of G using at most k colors such that each vertex has at most Δ⁎ neighbors in the same color class. Defective Coloring is a generalization of Graph Coloring with Δ⁎=0. The optimization variant of this problem, which aims to find the minimum number of colors k, is known to be NP-hard even for split graphs and cographs. Belmonte, Lampis, and Mitsou (SIDMA 2020) showed that Defective Coloring is W[1]-hard when parameterized by feedback vertex set or by treedepth, which implies W[1]-hardness for path-width and treewidth parameters. The problem is W[1]-hard parameterized by modular-width or clique-width as Defective Coloring is NP-hard on cographs. They asked whether Defective Coloring is fixed-parameter tractable (FPT) when parameterized by modular-width, neighborhood diversity or clique-width combined with either k or Δ⁎. In an effort to address the question concerning modular-width, this study investigates the parameters neighborhood diversity and twin-cover, which are special cases of modular-width. We show that Defective Coloring is FPT when parameterized by twin-cover, distance to disjoint paths, or the combined parameters neighborhood diversity and k. The latter implies an FPT algorithm for complete-d-partite graphs, a subclass of cographs, parameterized by d. We present an algorithm for graphs with bounded distance to d-degree. We also present a 1-additive approximation algorithm for split graphs.