Extremal Independent Set Reconfiguration

01/05/2023

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by   Nicolas Bousquet, et al.

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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⌊ k/3 ⌋ lower bound.