On the parameterized complexity of happy vertex coloring

Let G be a graph, and c: V (G) → [k] be a coloring of vertices in G. A vertex u ∈ V (G) is happy with respect to c if for all v ∈ NG(u), we have c(u) = c(v), i.e. all the neighbors of u have color same as that of u. The problem Maximum Happy Vertices takes as an input a graph G, an integer k, a vertex subset S ⊆ V (G), and a (partial) coloring c: S → [k] of vertices in S. The goal is to find a coloring c: V (G) → [k] such that c|S = c, i.e. c extends the partial coloring c to a coloring of vertices in G and the number of happy vertices in G is maximized. For the family of trees, Aravind et al. [1] gave a linear time algorithm for Maximum Happy Vertices for every fixed k, along with the edge variant of the problem. As an open problem, they stated whether Maximum Happy Vertices admits a linear time algorithm on graphs of bounded (constant) treewidth for every fixed k. In this paper, we study the problem Maximum Happy Vertices for graphs of bounded treewidth and give a linear time algorithm for every fixed k and (constant) treewidth of the graph. We also study the problem Maximum Happy Vertices with a different parameterization, which we call Happy Vertex Coloring. The problem Happy Vertex Coloring takes as an input a graph G, integers l and k, a vertex subset S ⊆ V (G), and a coloring c: S → [k]. The goal is to decide if there exist a coloring c: V (G) → [k] such that c|S = c and |H| = l, where H is the set of happy vertices in G with respect to c. We show that Happy Vertex Coloring is W[1]-hard when parameterized by l. We also give a kernel for Happy Vertex Coloring with O(k²l²) vertices.