The complexity of the defensive domination problem in special graph classes

A k-attack on a graph G is a set of k distinct vertices {a(1), ... , a(k)} which are said to be under attack. A k-attack A can be countered or defended by a subset of defender vertices X if and only if there exists an injective function f from A to X, such that either f(a(i)) = a(i) or (a(i), f(a(i))) is an edge of G, for all i, 1 <= i <= k. Given a graph G, a subset D of V is a k-defensive dominating set of G if and only if D can counter any k-attack in G. We consider the k-defensive dominating set problem.