Parameterized Algorithms for Locally Minimal Defensive Alliance

A set $D$ of vertices of a graph is a 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 locally minimal defensive alliance of maximum size. 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 and Connected Locally Minimal Defensive Alliance are fixed-parameter tractable (FPT) when parameterized by the solution size and $\Delta$, (2) Locally Minimal Defensive Alliance on the graphs of minimum degree at least 2, admits a kernel with at most ${f(k)}^{2k^2+4k}$ vertices for some computable function $f(k)$. In particular, we prove that the problem on triangle-free graphs of minimum degree at least 2, admits a kernel with at most $k^{\mathcal{O}(k^2)}$ vertices, where as the problem on planar graphs of minimum degree at least 2, admits a kernel with at most $k^{\mathcal{O}(k^4)}$ vertices. We also prove that (3) Locally Minimal DA Extension is NP-complete.