Network redesign problems ask to modify the edges of a given graph to satisfy some 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 temporal graph reconfiguration problem. As a starting point, we consider the Layered Connectivity Reconfiguration problem in which every snapshot of the temporal graph must remain connected throughout the reconfiguration. We provide insights into how bridges can be reconfigured into non-bridges based on their reachability partitions, which lets us identify any edge as either changeable or unchangeable. From this we construct a polynomial-time algorithm that gives a valid reconfiguration sequence of length at most 2M^2 (where M is the number of temporal edges), or determines that reconfiguration is not possible. We also show that minimizing the length of the reconfiguration sequence is NP-hard via a reduction from vertex cover.

翻译：网络重构问题旨在修改给定图的边以满足特定性质。在时序图中，边仅在特定时刻处于活跃状态，有时我们仅被允许修改边的活跃时间。在实际应用中，我们甚至可能无法一次性完成所有必要的修改；变更必须在网络仍在运行时逐步实施，这意味着网络必须持续满足某些性质。为开启该领域的研究，我们提出了时序图重构问题。作为起点，我们考虑分层连通性重构问题，其中时序图的每个快照必须在整个重构过程中保持连通。我们深入探讨了如何基于桥的可达性划分将其重构为非桥结构，从而能够识别任意边为可变更或不可变更。基于此，我们构建了一个多项式时间算法，该算法可生成长度至多为2M^2（其中M为时序边数量）的有效重构序列，或判定重构不可行。此外，我们通过顶点覆盖问题的归约证明最小化重构序列长度是NP难问题。