Parameterizing Role Coloring on Forests

Abstract

A role coloring of a graph is an assignment of colors to the vertices such that if any two vertices are assigned the same color, then their neighborhood are assigned the same set of colors. In k-Role Coloring, we want to ask whether a given graph is role colorable by using exactly k colors.

Determining whether a graph admits a k-Role Coloring is a notoriously hard problem even for a fixed $k \ge 2$. It is known to be NP-complete for $k \ge 2$ on general graphs. For many hereditary graph classes like chordal graphs, planar graphs and split graphs, k-Role Coloring is NP-complete even when k is a constant. A recent result shows that k-Role Coloring is NP-complete for bipartite graphs when $k \ge 3$. The only known classes of graphs for which k-role coloring is polynomial time for any fixed k, are Cographs and Proper Interval graphs.

We consider the parameterized complexity of $(n-k)$-role coloring on n vertex graphs parameterized by k. This parameterization had interesting fixed-parameter tractable algorithms for the standard proper coloring [Chor et al. WG2004] and list-coloring variants [Banik et al. IWOCA 2019, Gutin et al. STACS 2020]. As our main results, we show that this parameterization for role coloring has

• an $n^{O(k)}$ algorithm on general graphs (putting the problem in the parameterized complexity class XP), and

• an $f(k)n^{O(1)}$ algorithm (placing the problem in the class FPT) on forests (here f is a computable exponential function).