Fault-Tolerant Distance Labeling for Planar Graphs

02/14/2021

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by   Aviv Bar-Natan, et al.

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In fault-tolerant distance labeling we wish to assign short labels to the vertices of a graph G such that from the labels of any three vertices u,v,f we can infer the u-to-v distance in the graph G∖{f}. We show that any directed weighted planar graph (and in fact any graph in a graph family with O(√(n))-size separators, such as minor-free graphs) admits fault-tolerant distance labels of size O(n^2/3). We extend these labels in a way that allows us to also count the number of shortest paths, and provide additional upper and lower bounds for labels and oracles for counting shortest paths.