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arXiv · 2301.02020

Extremal Independent Set Reconfiguration

Nicolas Bousquet·Bastien Durain·Théo Pierron·Stéphan Thomassé
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Abstract

The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on independent sets are widely studied: for example, it is well known that an $n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is tight). This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size $k$ among all $n$-vertex graphs. We give a tight bound for $k=2$. We also provide a subquadratic upper bound (using the hypergraph removal lemma) as well as an almost tight construction for $k=3$. We generalize our results for larger values of $k$ by proving an $n^{2\lfloor k/3 \rfloor}$ lower bound.

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Nicolas Bousquet, Bastien Durain, Théo Pierron, Stéphan Thomassé. 2023-01-05. Extremal Independent Set Reconfiguration. https://arxiv.org/abs/2301.02020

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