Finding the Most Vital Edges for Shortest Paths Algorithms and Complexity for Special Graph Classes

We study the NP-hard problem of finding the most vital edges for shortest paths between two terminals of an undirected graph, that are a small number of edges whose deletion increases the distance between the terminals significantly. The associated decision problem asks whether a given edge deletion budget is sufficient to achieve the desired length increase. We study this decision problem as well as three variants, where edges are assigned a length or a deletion cost or both, in the context of selected graph classes. We show that the problem remains NP-complete on all superclasses of complete graphs when either edge weight is non-uniform. In the case where both are uniform, we give linear-time decision algorithms for complete and threshold graphs and conjecture the correctness of a polynomial-time algorithm that decides the problem on proper interval graphs. We also look into multigraphs and provide two FPT algorithms for the two-terminal series-parallel graphs that have polynomial running time if at least one of the edge weights is uniform.

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