Separating free-start pursuit models with and without waiting

Establish whether the free-start pursuit model in which the evader may start before the search but cannot wait at vertices is distinct from the waiting-free-start model in which the evader may both start early and wait at vertices.

Background

The paper studies two evader capabilities separately and jointly: the evader may start an unknown number of time steps before the pursuer begins searching, and the evader may wait at vertices. The authors prove an exponential-time winning strategy for the combined waiting-free-start model and note that the same strategy also handles early starts when waiting is forbidden.

What remains unresolved is whether forbidding waiting changes the model’s solvability or optimal search-time behavior. The paper explicitly leaves open whether the early-start-with-forced-movement and early-start-with-waiting settings can be separated.

References

The same strategy applies to the setting where the evader starts early but cannot wait, and we leave it as an open question whether these settings can be separated.

— Blindfolded pursuit with delays of your choice  (2608.27347 - Schürenberg et al., 27 Aug 2026) in Section 4, “Starting late”, opening paragraph

We conjecture that these strategies are the shortest possible, but know this (from complete enumeration) only for a maximum delay of nine.

— Blindfolded pursuit with delays of your choice  (2608.27347 - Schürenberg et al., 27 Aug 2026) in Appendix, Section 7, “Short strategies for the most general setting”