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 its closed neighbours are in $S$, that is, $|N_G[v] \cap S| \geq |N_G[v] \setminus S|$. The decision version of the problem is known to be NP-Complete even when restricted to split and bipartite graphs. The problem is \textit{fixed-parameter tractable} for the parameters solution size, vertex cover number and neighbourhood diversity. For the parameters treewidth and feedback vertex set number, the problem is W[1]-hard. \\ \hspace*{2em} In this paper, we study the defensive alliance problem for graphs with bounded degree. We show that the problem is \textit{polynomial-time solvable} on graphs with maximum degree at most 5 and NP-Complete on graphs with maximum degree 6. This rules out the fixed-parameter tractability of the problem for the parameter maximum degree of the graph. We also consider the problem from the standpoint of parameterized complexity. We provide an FPT algorithm using the Integer Linear Programming approach for the parameter distance to clique. We also answer an open question posed in \cite{AG2} by providing an FPT algorithm for the parameter twin cover.

翻译：给定图 $G = (V, E)$，非空集合 $S \subseteq V$ 称为一个防御联盟，若对于每个顶点 $v \in S$，其闭邻域中的多数顶点属于 $S$，即 $|N_G[v] \cap S| \geq |N_G[v] \setminus S|$。该问题的判定版本即使在限制于分裂图与二分图时，已知为 NP-完全问题。问题对于参数解大小、顶点覆盖数及邻域多样性是\textit{固定参数可解}的。而对于参数树宽与反馈顶点集数，问题为 W[1]-困难。 本文研究有界度图上的防御联盟问题。我们证明当图的最大度不超过 5 时，问题可在\textit{多项式时间}内求解，而当最大度为 6 时则为 NP-完全问题。这一结果排除了问题关于参数最大度的固定参数可解性。我们还从参数化复杂性的角度考虑该问题，提出一种基于整数线性规划方法的 FPT 算法，其参数为到团的距离。此外，我们回答了 \cite{AG2} 中提出的开放问题，给出了参数为双胞胎覆盖的 FPT 算法。