On Structural Parameterizations of Star Coloring

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 χ_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.