Parameterized algorithms for the Happy Set problem - Asahiro, Yuichi; Eto, Hiroshi; Hanaka, Tesshu; Lin, Guohui; Miyano, Eiji; Terabaru, Ippei (Discrete Applied Mathematics, 2021)
In this paper we study the parameterized complexity for the Maximum Happy Set problem (MaxHS): For an undirected graph G=(V,E) and a subset S⊆V of vertices, a vertex v is happy if v and all its neighbors are in S; otherwise unhappy. Given an undirected graph G=(V,E) and an integer k, the goal of MaxHS is to find a subset S⊆V of k vertices such that the number of happy vertices is maximized. In this paper we first show that MaxHS is W[1]-hard with respect to k even if the input graph is a split graph. Then, we prove the fixed-parameter tractability of MaxHS when parameterized by tree-width, by clique-width plus k, by neighborhood diversity, or by cluster deletion number.