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In this draft we prove an interesting structural property related to theproblem of computing {\em all the best swap edges} of a {\em tree spanner} inunweighted graphs. Previous papers show that the maximum stretch factor of thetree where a failing edge is temporarily swapped with any other available edgethat reconnects the tree depends only on the {\em critical edge}. However, inprinciple, each of the $O(n^2)$ swap edges, where $n$ is the number of verticesof the tree, may have its own critical edge. In this draft we show that thereare at most 6 critical edges, i.e., each tree edge $e$ has a {\em critical set}of size at most 6 such that, a critical edge of each swap edge of $e$ iscontained in the critical set.

An Interesting Structural Property Related to the Problem of Computing All the Best Swap Edges of a Tree Spanner in Unweighted Graphs