Temporal Graph Reconfiguration for Always-Connected Graphs

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.