Network redesign problems ask for modifications to the edges of a given graph to satisfy certain properties. In temporal graphs, where edges are only active at certain times, we are sometimes only allowed to modify when the edges are going to be active. In practice, we might not even be able to perform all of the necessary modifications at once; changes must be applied step-by-step while the network is still in operation, meaning that the network must continue to satisfy some properties. To initiate a study in this area, we introduce the class of temporal graph reconfiguration problems. As a starting point, we consider the Layered Connectivity Reconfiguration (LCR) problem: Given two always-connected temporal graphs G1 and G2, determine if it is possible to transform G1 into G2 by changing the time at which a single temporal edge is active in each step, such that every intermediate temporal graph is always-connected. We provide a dynamic programming algorithm for the LCR problem. We also show that finding the shortest reconfiguration sequence between two temporal graphs is APX-hard. Additionally, we show that the LCR problem is equivalent to the Spanning Tree Sequence Reconfiguration (STSR) problem introduced by Hanaka et al. Therefore, our results also answer the two open questions presented by the authors: (i) find a simpler algorithm for the STSR problem, (ii) show that the STSR problem is inapproximable up to some factor.

翻译：网络重设计问题要求对给定图的边进行修改以满足特定属性。在时态图中，边仅在特定时间激活，我们有时仅允许修改边的激活时间。实际应用中，我们甚至可能无法一次性完成所有必要修改；变更必须在网络运行期间逐步实施，这意味着网络必须持续满足某些属性。为启动该领域的研究，我们引入时态图重构问题类。作为起点，我们考虑分层连通性重构（LCR）问题：给定两个始终连通的时态图G1和G2，判断是否可以通过每一步修改单条时态边的激活时间将G1转化为G2，使得每个中间时态图始终保持连通。我们为LCR问题提出了一种动态规划算法。同时证明，寻找两个时态图间的最短重构序列是APX-hard的。此外，我们证明LCR问题等价于Hanaka等人提出的生成树序列重构（STSR）问题。因此，我们的结果也回答了作者提出的两个开放问题：（i）为STSR问题找到更简单的算法；（ii）证明STSR问题在特定因子下不可近似。