A tree t-spanner T of a graph G is a spanning tree with the property that the distance between any pair of nodes in T is at most t times the distance in G. If an edge e in T fails (i.e., is removed from G and T), the tree breaks into two subtrees \(T_+\) and \(T_-\) . Let \(E_X\) denote the set of edges in G that reconnect \(T_+\) and \(T_-\) . Every edge \(f\in E_X\) is a potential swap edge for e that can be used to repair the tree spanner. An edge in \(E_X\) that has the largest stretch among all edges in \(E_X\) when f is used to reconnect \(T_+\) and \(T_-\) is called a critical edge. In this paper, we show that, for every edge e 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 e. 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.
