Linear upper bounds for independent-set reconfiguration with arbitrary token number

Determine whether the maximum token-sliding reconfiguration distance \(\mu_k(n)\) for independent sets of size \(k\) in graphs on \(n\) vertices is bounded by \(O(n)\) for every positive integer \(k\), particularly for values \(k\geq 3\) not resolved by the two-token result.

Background

The paper defines μk(n)\mu_k(n) as the maximum, over all graphs on nn vertices and all mutually reachable independent sets of size kk, of the minimum number of token-sliding moves needed to transform one set into the other. A general counting argument gives only an O(nk)O(n^k) upper bound.

Bri-anski, Felsner, Hodor, and Micek posed the question of whether a linear bound holds for every token number kk. The paper establishes the bound μ2(n)≤4n\mu_2(n)\leq 4n, thereby resolving the case of two tokens, but does not resolve the general case.

References

Is $\mu_k(n)\leq O(n)$ for every $k\in\mathbb{N}$? In fact, they note that the question of whether there is a linear upper bound for $\mu_k(n)$ is open even for $k=2$.

— A linear upper bound on the number of moves required for independent set reconfiguration with two sliding tokens  (2608.13130 - M. et al., 13 Aug 2026) in Section 1, Introduction