A set $D$ of vertices of a graph is a \emph{defensive alliance} if, for each element of $D$, the majority of its neighbours are in $D$. We consider the notion of local minimality in this paper. We are interested in finding a locally minimal defensive alliance of maximum size. In Locally Minimal Defensive Alliance problem, given an undirected graph $G$, a positive integer $k$, the question is to check whether $G$ has a locally minimal defensive alliance of size at least $k$. This problem is known to be NP-hard, but its parameterized complexity remains open until now. We enhance our understanding of the problem from the viewpoint of parameterized complexity. The main results of the paper are the following: (1) Locally Minimal Defensive Alliance restricted to the graphs of minimum degree at least 2 is fixed-parameter tractable (FPT) when parameterized by the combined parameters solution size $k$, and maximum degree $\Delta$ of the input graph, (2) Locally Minimal Defensive Alliance on the graphs of minimum degree at least 2, admits a kernel with at most $k^{k^{\mathcal{O}(k)}}$ vertices. In particular, the problem parameterized by $k$ restricted to $C_3$-free and $C_4$-free graphs of minimum degree at least 2, admits a kernel with at most $k^{\mathcal{O}(k)}$ vertices. Moreover, we prove that the problem on planar graphs of minimum degree at least 2, admits an FPT algorithm with running time $\mathcal{O}^{*}(k^{2^{\mathcal{O}(\sqrt{k})}})$. Finally, we prove that (4) Locally Minimal Defensive Alliance Extension is NP-complete.
翻译:在本地最小防御联盟中, 一个不方向的图形$G$, 一个正整数$K, 问题在于检查$G$是否具有本地最低防御联盟, 规模至少为$美元。 这个问题已知是硬的, 但它的参数化复杂性至今仍然开放。 我们从参数化复杂度的角度考虑本地最小值的概念。 我们感兴趣的是找到本地最小防御性最大大小的联盟。 在本地最小防御性联盟中, 一个非方向化图形$G$, 一个正整数 美元, 问题在于检查$G$是否具有本地最低防御联盟, 规模至少为$@rk。 问题在于检查是否本地最小值的防御联盟, 最小值为NP- 硬值, 但它的参数化复杂性一直到现在为止。 我们从参数化复杂度的角度增进了对问题的理解。 本地最小值 最小值的联盟, 最低值为 最低值为本地值 。