A Tight Bound on the Size of a Smallest Critical Set for Each Failed Edge in a Tree Spanner

A tree [[EQUATION]]-spanner [[EQUATION]] of an edge-weighted graph [[EQUATION]] is a spanning tree with the property that the distance between any pair of nodes in [[EQUATION]] is at most [[EQUATION]] times the distance in [[EQUATION]]. If an edge [[EQUATION]] in [[EQUATION]] fails, the tree breaks into two subtrees [[EQUATION]] and [[EQUATION]]. Let [[EQUATION]] denote the set of edges in [[EQUATION]] that reconnect [[EQUATION]] and [[EQUATION]]. Every edge [[EQUATION]] is a potential swap edge for [[EQUATION]] that can be used to repair the tree spanner. An edge in [[EQUATION]] that has the largest stretch among all edges in [[EQUATION]] when [[EQUATION]] is used to reconnect [[EQUATION]] and [[EQUATION]] is called a critical edge. In this paper, we show that, for every edge [[EQUATION]] that fails in a tree spanner of an unweighted graph, there is always a set of at most four edges that contains at least one critical edge for every potential swap edge of [[EQUATION]]. The proof relies on studying the endpoints of a diametrical path in a tree with changing edge weights and may be of independent interest. We also show that there are instances where the smallest critical set has size four.