Extremal Independent Set Reconfiguration

TL;DR

This paper studies the maximum length of reconfiguration sequences between independent sets in graphs, providing tight bounds for small sizes and general bounds for larger sizes, advancing understanding of independent set transformations.

Contribution

It offers new bounds on reconfiguration sequence lengths for independent sets, including tight bounds for size 2, subquadratic bounds for size 3, and general lower bounds for larger sizes.

Findings

Tight bound for k=2 reconfiguration sequences.

Subquadratic upper bound for k=3 reconfiguration sequences.

Lower bound of n^{2*floor(k/3)} for larger k.

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.