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

TL;DR

This paper proves a structural property that limits the number of critical edges to at most six for all swap edges in a tree spanner of an unweighted graph, simplifying the computation of optimal swap edges.

Contribution

It establishes that each tree edge has a critical set of at most six edges, reducing the complexity of finding all best swap edges in unweighted graphs.

Findings

Maximum of 6 critical edges per tree edge.

Reduces complexity in computing swap edges.

Provides structural insight into tree spanners.

Abstract

In this draft we prove an interesting structural property related to the problem of computing {\em all the best swap edges} of a {\em tree spanner} in unweighted graphs. Previous papers show that the maximum stretch factor of the tree where a failing edge is temporarily swapped with any other available edge that reconnects the tree depends only on the {\em critical edge}. However, in principle, each of the $O(n^2)$ swap edges, where $n$ is the number of vertices of the tree, may have its own critical edge. In this draft we show that there are 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 $e$ is contained in the critical set.