Shortest paths between shortest paths and independent sets

TL;DR

This paper investigates the complexity of reconfiguring shortest paths and independent sets in graphs, revealing exponential bounds, NP-hardness, and polynomial cases in specific graph classes, while establishing relationships between these reconfiguration problems.

Contribution

It demonstrates the NP-hardness of shortest path reconfiguration, explores the connection between shortest path and independent set reconfiguration, and provides polynomial algorithms for certain graph classes.

Findings

Shortest reconfiguration sequences can be exponential in size.

NP-hardness of computing shortest reconfiguration sequences.

Polynomial algorithms for even-hole-free and P4-free graphs.

Abstract

We study problems of reconfiguration of shortest paths in graphs. We prove that the shortest reconfiguration sequence can be exponential in the size of the graph and that it is NP-hard to compute the shortest reconfiguration sequence even when we know that the sequence has polynomial length. Moreover, we also study reconfiguration of independent sets in three different models and analyze relationships between these models, observing that shortest path reconfiguration is a special case of independent set reconfiguration in perfect graphs, under any of the three models. Finally, we give polynomial results for restricted classes of graphs (even-hole-free and $P_4$-free graphs).