On the tractability of defensive alliance problem

Given a graph $G=(V,E)$, a non-empty set $S\subseteq V$ is a defensive alliance if, for every vertex $v\in S$, the majority of vertices in its closed neighbourhood belong to $S$; that is, $|N_G\left[v\right]\cap S|\ge |N_G\left[v\right]\setminus S|$. The Defensive Alliance problem (Defensive Alliance) asks for a defensive alliance of minimum cardinality. The decision version of the problem is known to be NP-complete even when restricted to split graphs and bipartite graphs. From a parameterized complexity perspective, the Defensive Alliance is known to be fixed-parameter tractable (FPT) when parameterized by the solution size, the vertex cover number, or the neighbourhood diversity of the input graph. In contrast, the problem is W[1]-hard when parameterized by the treewidth or the feedback vertex set number.

In this paper, we investigate the complexity of the Defensive Alliance on bounded degree graphs. We prove that the problem is polynomial-time solvable on graphs with maximum degree at most 5 but becomes NP-complete when the maximum degree is 6. This result rules out fixed-parameter tractability with respect to the maximum degree. Additionally, we analyse the problem from the perspective of parameterized complexity and present an FPT algorithm parameterized by twin cover number, thereby resolving an open question posed in Gaikwad and Maity (2022).