Characterization of graphs attaining the radius upper bound

Characterize connected graphs G whose radius capture number satisfies rc(G) = rad(G) − 1.

Background

The paper proves the universal upper bound rc(G) ≤ rad(G) − 1 for every connected graph G. It then gives a sufficient condition guaranteeing equality, namely a condition describing how vertices at distance equal to the graph radius can be matched after one-step moves. The authors explicitly note that it is unclear whether this sufficient condition is also necessary and pose the broader characterization problem of determining all connected graphs that attain the upper bound.