Characterization of graphs equating capture and inference numbers

Characterize the graphs G for which the limited-visibility capture number cop′_l(G) equals the inference number ded¹_l(G), for a given positive visibility parameter l.

Background

The paper introduces a graph-cleaning formulation in which cops attempt to prevent gas from occupying vertices. The inference number ded¹_l(G) is the minimum number of visibility-l cops needed to reduce the set of possible robber locations to at most one vertex, whereas cop′_l(G) is the minimum number needed to clean the entire graph and thereby guarantee seeing or capturing the robber.

The paper proves that cop′_l(G)−ded¹_l(G) ≤ 1, and gives the cycle C₅ as an example where the difference is exactly one. It then identifies the structural characterization of graphs for which no additional cop is needed as an open problem.

References

The graph $C_5$ appears to be more than just a simple example illustrating the difference between being able to see the robber and knowing their location. Indeed, if the reader starts to investigate the following open problem (particularly with $l=1$), then they will see it cropping up frequently, but it is not clear exactly what is going on.

\begin{problem} Characterise the graphs $G$ for which $\cop'_l(G)=\ded1_l(G)$. \end{problem}

— Seeing is not believing in limited visibility cops and robbers  (2507.00941 - Bašić et al., 1 Jul 2025) in Section 3, subsection immediately following Corollary