Characterization of bipartite graphs for larger capture radii

Characterize bipartite graphs belonging to CWRC(k), equivalently bipartite graphs whose radius capture number is at most k, for every integer k ≥ 2.

Background

Prior work cited in the paper characterizes bipartite graphs in CWRC(1), the class of graphs in which one cop can win with radius of capture 1. The authors explicitly ask whether an analogous characterization is possible for all larger capture radii k ≥ 2. This problem concerns extending the known radius-one structural characterization to the general bipartite case.