On Structural Parameterizations of Star Coloring

TL;DR

This paper explores the computational complexity of star coloring in graphs, demonstrating fixed-parameter tractability for certain graph parameters despite the problem's NP-completeness in general.

Contribution

It introduces a parameterized complexity analysis of the star coloring problem, identifying conditions under which it becomes fixed-parameter tractable.

Findings

Star coloring is NP-complete on bipartite graphs.

The problem is fixed-parameter tractable with respect to neighborhood diversity.

It is also fixed-parameter tractable when parameterized by twin-cover and combined clique-width and color count.

Abstract

A Star Coloring of a graph G is a proper vertex coloring such that every path on four vertices uses at least three distinct colors. The minimum number of colors required for such a star coloring of G is called star chromatic number, denoted by \chi_s(G). Given a graph G and a positive integer k, the STAR COLORING PROBLEM asks whether $G$ has a star coloring using at most k colors. This problem is NP-complete even on restricted graph classes such as bipartite graphs. In this paper, we initiate a study of STAR COLORING from the parameterized complexity perspective. We show that STAR COLORING is fixed-parameter tractable when parameterized by (a) neighborhood diversity, (b) twin-cover, and (c) the combined parameters clique-width and the number of colors.