Construct an optimal-length dominating pair efficiently

Construct, in linear time for an arbitrary graph, a k-dominating pair whose domination radius k is at most the graph’s path-length.

Background

A k-dominating pair is a pair of vertices such that every path joining the pair is a k-dominating path. The paper explains that a minimum-radius dominating pair can be found in polynomial time, while a known linear-time algorithm returns a pair with radius at most twice the path-length.

The authors explicitly state that they do not know how to achieve the stronger bound equal to path-length in linear time. Thus, the unresolved issue is an algorithmic improvement from the existing factor-two guarantee to an exact path-length guarantee.