Globally Minimal Defensive Alliances: A Parameterized Perspective
Ajinkya Gaikwad, Soumen Maity

TL;DR
This paper investigates the computational complexity of finding maximum size globally minimal defensive alliances in graphs, revealing fixed-parameter tractability for certain parameters and hardness results for others.
Contribution
It establishes fixed-parameter tractability of the problem with respect to neighborhood diversity and vertex cover, and proves W[1]-hardness for several structural parameters.
Findings
FPT for neighborhood diversity and vertex cover
No polynomial compression unless coNP ⊆ NP/poly
W[1]-hard for feedback vertex set, pathwidth, treewidth, treedepth
Abstract
A defensive alliance in an undirected graph is a non-empty set of vertices satisfying the condition that every vertex has at least as many neighbours (including itself) in as it has in . We consider the notion of global minimality in this paper. We are interested in globally 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 by showing that the Globally Minimal Defensive Alliance problem is FPT parameterized by the neighbourhood diversity of the input graph. The result for neighborhood diversity implies that the problem is FPT parameterized by vertex cover number also. We prove that the problem parameterized by the vertex cover number of the input graph does not admit a…
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Taxonomy
TopicsAdvanced Graph Theory Research · Formal Methods in Verification · Cyclopropane Reaction Mechanisms
