Bounding iterations in the reducing-adhesion carving phase when few carvable vertices remain

Establish that the carving-based procedure in the reducing adhesion algorithm terminates within O(√n) iterations when the set Q of (X1, T, 2k)-carvable vertices has size at most √n by proving that successive carvings do not introduce a large number of new (X1, T, 2k)-carvable vertices. Specifically, given the algorithmic case where |Q| ≤ √n in the "Reducing Adhesion Fast" strategy, determine whether the total number of iterations can be bounded by a function of |Q| (e.g., O(√n)), despite the possibility that carving adds new vertices into T and thereby creates additional carvable vertices.

Background

The paper’s reducing-adhesion subroutine iteratively carves the terminal set T along connected lean (X1, T, k')-witnesses to decrease T \ X1 while maintaining unbreakability and bounded adhesion. A key structural lemma shows that the absence of carvable vertices implies stronger unbreakability, enabling progress.

To accelerate the process, the authors use a disjoint-witness cover to carve away a large fraction of carvable vertices when many remain. However, in the regime where the set Q of carvable vertices becomes small (|Q| ≤ √n), they cannot bound the number of remaining iterations, because each carve may introduce new vertices into T, creating additional carvable vertices even though |T \ X1| decreases.

A proof that the number of iterations is bounded by O(√n) (or a similar function of |Q|) would strengthen the runtime guarantees by eliminating the need for multi-level thresholds and directly controlling the iteration count in the low-|Q| regime.