Approximation Algorithm and Hardness Results for Defensive Domination in Graphs

Abstract

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 is considered to be under attack. A set $D \subseteq V$ can defend or counter the attack A on G if there exists a one to one function $f: A \longmapsto D$, such that either $f(u) =u$ or there is an edge between u and it’s 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 $D \subseteq V$ 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 graphs and threshold graphs for any k. In this paper, we mainly focus 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 give 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 $k>0$.