Parameterized Algorithms for Locally Minimal Defensive Alliance
Ajinkya Gaikwad, Soumen Maity, Saket Saurabh

TL;DR
This paper investigates the parameterized complexity of finding large locally minimal defensive alliances in graphs, providing fixed-parameter tractability results, kernel bounds, and NP-completeness for related extension problems.
Contribution
It introduces new fixed-parameter algorithms, kernel bounds, and complexity results for the Locally Minimal Defensive Alliance problem in various graph classes.
Findings
FPT algorithm for graphs with minimum degree at least 2 using parameters solution size and maximum degree
Kernel with at most $k^{k^{O(k)}}$ vertices for the problem on certain graphs
NP-completeness of the Locally Minimal Defensive Alliance Extension problem
Abstract
A set of vertices of a graph is a \emph{defensive alliance} if, for each element of , the majority of its neighbours are in . 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 , a positive integer , the question is to check whether has a locally minimal defensive alliance of size at least . 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…
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Taxonomy
TopicsAdvanced Graph Theory Research · Synthetic Organic Chemistry Methods · Cholinesterase and Neurodegenerative Diseases
