Fault-tolerant distance labeling for planar graphs

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.