Sublog-squared product of hop-diameter and treewidth for tree shortcuttings

Determine whether there exists, for every n, a shortcutting of an n-vertex tree with hop-diameter k and treewidth t such that the product k · t = o((log log n)^2); furthermore, ascertain whether such a shortcutting can be achieved with constant hop-diameter k.

Background

Filtser and Le established a construction of tree shortcuttings achieving hop-diameter O(log log n) and treewidth O(log log n), which underpins low-treewidth embeddings of planar graphs into low-treewidth graphs. Improving the product k * t would directly strengthen bounds in such embeddings. The authors explicitly note that the following question remained open in that prior work.