On the structural parameterized complexity of defective coloring

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...

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Vydáno v:Journal of computer and system sciences Ročník 157; s. 103736
Hlavní autoři: Bhyravarapu, Sriram, Kumar, Pankaj, Saurabh, Saket
Médium: Journal Article
Jazyk:angličtina
Vydáno: Elsevier Inc 01.05.2026
Témata:
Bounded degree graphs
Defective coloring
Neighborhood diversity
Parameterized complexity
Split graphs
Twin-cover
Twin-cover
Split graphs
Neighborhood diversity
Defective coloring
Parameterized complexity
Bounded degree graphs
ISSN:0022-0000
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Shrnutí: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.
ISSN:0022-0000
DOI:10.1016/j.jcss.2025.103736