2020/03/10 by Bliznets, Ivan, Sagunov, Danil
#Data Structures and Algorithms (cs.DS) #F.2.0 #FOS: Computer and information sciences
paper · doi:10.48550/arxiv.2003.04605
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(ℓ n2) algorithm for MHV on interval graphs.