Tighten bounds on the cop number of string graphs

Determine whether the maximum cop number over the class of string graphs can be reduced below 13 by developing a winning strategy that uses fewer than 13 cops, or construct a specific string graph whose cop number is at least 4 in order to raise the known lower bound and narrow the current interval 3 ≤ c(𝒮) ≤ 13.

Background

The paper improves the best known upper bound on the cop number for string graphs from 15 to 13 by introducing a new guarding technique for convex paths and combining it with isometric-path strategies. While the upper bound is lowered, the authors point out that there is presently no explicit example of a string graph requiring four cops; hence the class-wide cop number is only known to lie between 3 and 13.

Consequently, the authors explicitly pose the problem of tightening this range. This can be achieved either by further reducing the upper bound via improved strategies or by increasing the lower bound through explicit construction of a string graph with cop number at least four.