Perfect Domination with Forbidden Vertices on lk-starlike Graphs and Trees

Let G(V, E) be a graph with n-vertex-set V and m-edge-set E. Meanwhile, each vertex v is associated with a nonnegative weight W(v). Given a subset F of V, this paper studies the problem of finding a perfect dominating set D such that δ(D) = ∑ is minimized under the restriction that D ⊆ (V F). The vertices in F are called forbidden vertices. We first define a new class of graphs, called l ∈D v D W ) ( k-starlike graphs, k ≥ 1, which is a super class of the class of starlike graphs. Then, we show that the problem is NP-Hard on lk-starlike graphs, k ≥ 2, but linear-time solvable when k = 1. Finally, an O(n)-time algorithm for the problem on trees is designed.