Construct an exact minimum path-length dominating pair efficiently

Develop a linear-time algorithm that, for an arbitrary graph G, finds a k-dominating pair with k at most the path-length of G.

Background

The paper notes that a minimum-k dominating pair can be found in O(n^3 log n) time, while an existing linear-time algorithm finds a dominating pair with k at most twice the path-length. The authors explicitly state that they do not know how to obtain a linear-time algorithm achieving the stronger bound k at most the path-length itself.