More on the complexity of defensive domination in graphs

In a graph G=(V,E), a non-empty set A of k distinct vertices, is called a k-attack on G. The vertices in the set A are considered to be under attack. A set D⊆V can defend or counter the attack A on G if there exists a one-to-one function f:A⟼D, such that either f(u)=u or there is an edge between u, and its image f(u), in G. A set D is called a k-defensive dominating set if it defends against any k-attack on G. Given a graph G=(V,E), the minimum k-defensive domination problem requires us to compute a minimum cardinality k-defensive dominating set of G. When k is not fixed, it is co-NP-hard to decide if D⊆V is a k-defensive dominating set. However, when k is fixed, the decision version of the problem is NP-complete for general graphs. On the positive side, the problem can be solved in linear time when restricted to paths, cycles, co-chain, and threshold graphs for any k. This paper mainly focuses on the problem when k>0 is fixed. We prove that the decision version of the problem remains NP-complete for bipartite graphs; this answers a question asked by Ekim et al. (Discrete Math. 343 (2) (2020)). We establish a lower and upper bound on the approximation ratio for the problem. Further, we show that the minimum k-defensive domination problem is APX-complete for bounded degree graphs. On the positive side, we show that the problem is efficiently solvable for complete bipartite graphs for any k>0. Towards the end, we study a relationship between the defensive domination number and another well-studied domination parameter.