一个有趣的结构性质与计算无权图中树生成子的所有最佳交换边问题有关
An Interesting Structural Property Related to the Problem of Computing All the Best Swap Edges of a Tree Spanner in Unweighted Graphs
摘要：
在这个草案中，我们证明了一个有趣的结构性质有关的问题，计算{ \em所有最好的交换边}的{ \em树}在未加权图。以前的论文表明，树的最大拉伸因子，其中失败的边缘是暂时交换与任何其他可用的边缘，重新连接的树只依赖于{ \em临界边}。然而，原则上，每个$O(n^2)$交换边（其中$n$是树的顶点数）可以具有其自己的临界边。在这个草案中，我们表明，最多有6个临界边，即，每个树边$e$具有大小至多为6的{ \em临界集}，使得$e$的每个交换边的临界边包含在临界集中。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
