For k-chordal graphs it builds O(1)-round shortcuts of quality O(kD), and for diameter 3 and 4 it builds shortcuts of quality O~(n^{1/4}) and O~(n^{1/3}) that yield matching MST algorithms.
Garay, Shay Kutten, and David Peleg
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Low-Congestion Shortcut and Graph Parameters
For k-chordal graphs it builds O(1)-round shortcuts of quality O(kD), and for diameter 3 and 4 it builds shortcuts of quality O~(n^{1/4}) and O~(n^{1/3}) that yield matching MST algorithms.