Intermediate strong cop numbers in connected vertex-transitive graphs

Determine whether there exists a connected vertex-transitive graph G whose strong cop number satisfies \(\scop(G)\notin\{1,\infty\}\).

Background

The paper studies weak and strong cop numbers for infinite graphs. Its main theorem proves that every vertex-transitive graph containing a thick end has infinite weak cop number, and the resulting corollary shows that every connected vertex-transitive graph has weak cop number either 1 or infinity. Thus, the weak-cop-number alternative in the explicitly posed disjunctive question is resolved negatively by the paper.

The corresponding issue for the strong cop number is not resolved: the paper does not establish that every connected vertex-transitive graph has strong cop number 1 or infinity, nor does it provide a graph with an intermediate finite strong cop number. The unresolved problem is therefore to determine whether such a graph exists.