Parameterized Algorithms for the Happy Set Problem In this paper we introduce the Maximum Happy Set problem (MaxHS) and study its parameterized complexity: For an undirected graph (G  (V, E)) and a subset (Ssubseteq V) of vertices, a vertex v is happy if v and all its neighbors are in S; and otherwise unhappy. Given an undirected graph (G  (V, E)) and an integer k, the goal of MaxHS is to find a subset (Ssubseteq V) of k vertices such that the number of happy vertices is maximized. In this paper we first show that MaxHS is W1hard when parameterized by k. Then, we prove the fixedparameter tractability of MaxHS when parameterized by the treewidth, the cliquewidth and k, the neighborhood diversity, or the twincover number.

Subset feedback vertex set on graphs of bounded independent set size Abstract   The ( Weighted )  Subset Feedback Vertex Set  problem is a generalization of the classical  Feedback Vertex Set  problem and asks for a vertex set of minimum (weight) size that intersects all cycles containing a vertex of a predescribed set of vertices. Although  Subset Feedback Vertex Set  and  Feedback Vertex Set  exhibit different computational complexity on split graphs, no similar characterization is known on other classes of graphs. Towards the understanding of the complexity difference between the two problems, it is natural to study the importance of structural graph parameters. Here we consider graphs of bounded independent set number for which it is known that  Weighted Feedback Vertex Set  can be solved in polynomial time. We provide a dichotomy result with respect to the size xcexb1 of a maximum independent set. In particular we show that  Weighted Subset Feedback Vertex Set  can be solved in polynomial time for graphs with    xcexb1  xe2x89xa4  3   , whereas we prove that the problem remains NPhard for graphs with    xcexb1  xe2x89xa5  4   . Moreover, we show that the (unweighted)  Subset Feedback Vertex Set  problem can be solved in polynomial time on graphs of bounded independent set number by giving an algorithm with running time      n    O  (  xcexb1  )     . To complement our results, we demonstrate how our ideas can be extended to other terminal set problems on graphs of bounded independent set size.  Node Multiway Cut  is a terminal set problem that asks for a vertex set of minimum size that intersects all paths connecting any two terminals. Based on our findings for  Subset Feedback Vertex Set , we settle the complexity of  Node Multiway Cut  as well as its variants where nodes are weighted andor the terminals are deletable, for every value of the given independent set number.