Threshold Dominating Sets and an Improved Characterization of W[2]

The Threshold Dominating Set problem is that of determining for a graph G = (V, E) whether there is a subset V ⊆ V of size k, such that for each vertex v ∈ V there are at least r elements of the closed neighborhood N[v] that belong to V . We consider the complexity of the problem parameterized by the pair (k, r). It is trivial to observe that this is hard for W[2]. It can also be easily shown to belong to a natural extension W∗[2] of W[2] defined in terms of circuit families of depth bounded by a function of the parameter. We prove membership in W[2] and thus W[2]-completeness. Using this as a starting point, we prove that W∗[2] = W[2]. Key Words: parameterized complexity, dominating sets, threshold computation, satisfiability problems, boolean circuits.

• Type: Journal
• Year: 1998
• Where: TCS
• Authors: Rodney G. Downey, Michael R. Fellows
• Keywords: Circuit Families, Natural Extension W∗, TCS 1998, Theoretical Computer Science, Threshold Dominating Set