On approximating tree spanners that are breadth first search trees

There are known efficient algorithms that approximate the minimum t for which a graph admits a tree t-spanner with ratio O(log⁡n). Unless there is an algorithm that decides 3-SAT in quasi-polynomial time, it is impossible to approximate efficiently the minimum t for which a graph admits a tree t-spanner that is a breadth first search tree starting from a given vertex with ratio almost o(log⁡n).