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arXiv · 2609.27984

Parameterized Enumeration of Minimal Defensive Alliances

Henning Fernau·Kevin Mann·Arne Meier·Heribert Vollmer
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Abstract

In this paper, we consider the complexity of enumerating inclusion minimal defensive alliances. We present a polynomial-delay algorithm on graphs with maximum degree 5. We complement this result by proving that there is no output-polynomial algorithm on bipartite graphs with maximum degree 6 and degeneracy 2, unless P = NP. Furthermore, there is an FPT-delay algorithm when parameterized by neighborhood diversity. This result is the first exploit of a recently published enumeration algorithm for ILPs. By way of contrast, we prove that, for the parameter pathwidth, there is no FPT-delay algorithm for enumerating all inclusion minimal defensive alliances (unless FPT = W[1]). To the best of our knowledge, this is the first non-enumerability result using parameterized complexity for variations of Another/Next-problems.

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Henning Fernau, Kevin Mann, Arne Meier, Heribert Vollmer. 2026-09-23. Parameterized Enumeration of Minimal Defensive Alliances. https://arxiv.org/abs/2609.27984

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