Branch and recharge: Exact algorithms for generalized domination

Abstract

Let σ and ∂ be two sets of nonnegative integers. A vertex subset S ⊆ V of an undirected graph G = (V, E) is called a (σ, ∂)-dominating set of G if |N(v)∩S| ∈ σ for all v ∈ S and |N(v)∩S| ∈ ∂ for all v ∈ V\S. This notion introduced by Telle generalizes many dominationtype graph invariants. For many particular choices of σ and ∂ it is NP-complete to decide whether an input graph has a (σ, ∂)-dominating set. We show a general algorithm enumerating all (σ, ∂)-dominating sets of an input graph G in time O*(cn) for some c < 2 using only polynomial space, if σ is successor-free, i.e., it does not contain two consecutive integers, and either both σ and ∂ are finite, or one of them is finite and σ ∩ ∂ = Ø. Thus in this case one can find maximum and minimum (σ, ∂)-dominating sets in time o(2n). Our algorithm straightforwardly implies a non trivial upper bound cn with c < 2 for the number of (σ, ∂)-dominating sets in an n-vertex graph under the above conditions on σ and ∂. Finally, we also present algorithms to find maximum and minimum ({P}, {q})-dominating sets and to count the ({p}, {q})-dominating sets of a graph in time O*(2n/2). © Springer-Verlag Berlin Heidelberg 2007.