Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time, so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem.When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalize the problem to when at most k (for a fixed integer k ≥ 2) contiguous vertices on a shortest path can be changed at a time.

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Reconfiguring two shortest paths in a graph means modifying one shortest path to the other by changing one vertex at a time, so that all the intermediate paths are also shortest paths. This problem has several natural applications, namely: (a) revamping road networks, (b) rerouting data packets in a synchronous multiprocessing setting, (c) the shipping container stowage problem, and (d) the train marshalling problem. When modelled as graph problems, (a) is the most general case while (b), (c) and (d) are restrictions to different graph classes. We show that (a) is intractable, even for relaxed variants of the problem. For (b), (c) and (d), we present efficient algorithms to solve the respective problems. We also generalise the problem to when at most k (for some k >= 2) contiguous vertices on a shortest path can be changed at a time.

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It is NP-hard to determine the minimum number of branching vertices needed in a singlesource distance-preserving subgraph of an undirected graph. We show that this problem can be solved in polynomial time if the input graph is an interval graph.In earlier work, it was shown that every interval graph with k terminal vertices admits an all-pairs distance-preserving subgraph with O(k log k) branching vertices [GR17a]. We consider graphs that can be expressed as the strong product of two interval graphs, and present a polynomial time algorithm that takes such a graph with k terminals as input, and outputs an all-pairs distance-preserving subgraph of it with O(k 2 ) branching vertices. This bound is tight. * kshitij.gajjar@tifr.res.in † jaikumar@tifr.res.in