Parameterized algorithms for the Happy Set problem

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 subset of V of vertices, a vertex upsilon is happy if upsilon 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 subset of 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. (C) 2021 Elsevier B.V. All rights reserved.