On Structural Parameterizations of Happy Coloring, Empire Coloring and Boxicity

Distance parameters are extensively used to design efficient algorithms for many hard graph problems. They measure how far a graph is from belonging to some class of graphs. If a problem is tractable on a class of graphs, then distances to provide interesting parameterizations to that problem. For example, the parameter vertex cover measures the closeness of a graph to an edgeless graph. Many hard problems are tractable on graphs of small vertex cover. However, the parameter vertex cover is very restrictive in the sense that the class of graphs with bounded vertex cover is small. This significantly limits its usefulness in practical applications. In general, it is desirable to find tractable results for parameters such that the class of graphs with the parameter bounded should be as large as possible. In this spirit, we consider the parameter distance to threshold graphs, which are graphs that are both split graphs and cographs. It is a natural choice of an intermediate parameter between vertex cover and clique-width. In this paper, we give parameterized algorithms for some hard graph problems parameterized by the distance to threshold graphs. We show that Happy Coloring and Empire Coloring problems are fixed-parameter tractable when parameterized by the distance to threshold graphs. We also present an approximation algorithm to compute the Boxicity of a graph parameterized by the distance to threshold graphs.