More on the complexity of 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 $A$ are 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 $f:A⟼D$, such that either $f\left(u\right)=u$ or there is an edge between $u$, and its image $f\left(u\right)$, 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, 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.