Robber-monotonicity of bounded-speed lazy variants at speeds two and three

Determine whether the visible lazy and invisible lazy Cops-and-Robber copwidth variants with speeds 2 and 3 have the robber-monotonicity property.

Background

The paper proves that, for every finite speed s≥4, neither the visible lazy nor the invisible lazy variant has the robber-monotonicity property. The authors explain that their construction does not trivially extend to speeds 2 and 3, leaving those cases unresolved. They also note a possible connection between robber-monotonicity and the computational complexity of deciding the copwidth, but do not establish such a connection.

References

Our construction can not be trivially modified for the case $s\in {2,3}$ and it is questionable whether the theorem actually holds for these.

— Fast and Furious: A study on Monotonicity and Speed in Cops-and-Robber Games  (2502.15396 - Fluck et al., 21 Feb 2025) in Section 4, “Robber Monotonicity and Lazy Variants”