Vertex-Coloring with Star-Defects

TL;DR

This paper studies a specialized defective coloring where monochromatic components are star-shaped, providing algorithms for outerplanar graphs and NP-completeness results for more general graph classes.

Contribution

It introduces a new class of defective colorings with star-shaped components, offering linear-time algorithms for outerplanar graphs and complexity results for broader graph classes.

Findings

Linear-time algorithm for 2-coloring outerplanar graphs

Outerpath graphs always admit such a 2-coloring

NP-completeness for non-planar and planar graphs with bounded degree

Abstract

Defective coloring is a variant of traditional vertex-coloring, according to which adjacent vertices are allowed to have the same color, as long as the monochromatic components induced by the corresponding edges have a certain structure. Due to its important applications, as for example in the bipartisation of graphs, this type of coloring has been extensively studied, mainly with respect to the size, degree, and acyclicity of the monochromatic components. In this paper we focus on defective colorings in which the monochromatic components are acyclic and have small diameter, namely, they form stars. For outerplanar graphs, we give a linear-time algorithm to decide if such a defective coloring exists with two colors and, in the positive case, to construct one. Also, we prove that an outerpath (i.e., an outerplanar graph whose weak-dual is a path) always admits such a two-coloring. Finally, we present NP-completeness results for non-planar and planar graphs of bounded degree for the cases of two and three colors.