Maximizing Happiness in Graphs of Bounded Clique-Width

Abstract

Clique-width is one of the most important parameters that describes structural complexity of a graph. Probably, only treewidth is more studied graph width parameter. In this paper we study how clique-width influences the complexity of the Maximum Happy Vertices (MHV) and Maximum Happy Edges (MHE) problems. We answer a question of Choudhari and Reddy ’18 about parameterization by the distance to threshold graphs by showing that MHE is NP -complete on threshold graphs. Hence, it is not even in XP when parameterized by clique-width, since threshold graphs have clique-width at most two. As a complement for this result we provide a nO(ℓ·cw) algorithm for MHE, where ℓ is the number of colors and cw is the clique-width of the input graph. We also construct an FPT algorithm for MHV with running time O∗((ℓ+ 1 )O(cw)), where ℓ is the number of colors in the input. Additionally, we show O(ℓn2) algorithm for MHV on interval graphs.