A Note on the Complexity of Defensive Domination.

In a graph G, a k-attack A is any set of at most k vertices and ℓ-defense D is a set of at most ℓ vertices. We say that defense D counters attack A if each a ∈ A can be matched to a distinct defender d ∈ D with a equal to d or a adjacent to d in G. In the defensive domination problem, we are interested in deciding, for a graph G and positive integers k and ℓ given on input, if there exists an ℓ-defense that counters every possible k-attack on G. Defensive domination is a natural resource allocation problem and can be used to model network robustness and security, disaster response strategies, and redundancy designs. The defensive domination problem is naturally in the complexity class Σ^P₂. The problem was known to be NP-hard in general, and polynomial-time algorithms were found for some restricted graph classes. In this note, we prove that the defensive domination problem is Σ^P₂-complete. We also introduce a natural variant of the defensive domination problem in which the defense is allowed to be a multiset of vertices. This variant is also Σ^P₂-complete, but we show that it admits a polynomial-time algorithm in the class of interval graphs. A similar result was known for the original setting in the class of proper interval graphs.