The logarithmic localization conjecture for invisible agents on graphs

Let $G$ be a finite, simple, undirected, connected graph of order $n$. A mouse moves along the graph, and a cat probes vertices, receiving relative-distance information that determines the possible mouse positions. The cat can localize the mouse up to distance $d$ when it has a strategy guaranteeing that the set $M_i$ of possible mouse positions has graph radius at most $d$ at some finite time.

Logarithmic localization conjecture. The cat can localize the mouse up to distance $O(\log n)$ on $G$.

The conjecture extends the paper's constant-distance localization results for bounded-degree trees and square grids. The $O(\log n)$ scale is motivated by the fact that $O(\log n)$ bits suffice to identify one of $n$ vertices, while the mouse may move a comparable distance during the time needed to obtain that information.