\~Optimal Fault-Tolerant Reachability Labeling in Planar Graphs

TL;DR

This paper introduces an optimal fault-tolerant reachability labeling scheme for directed planar graphs, enabling quick determination of reachability after removing a vertex, with labels of near-constant size and query time.

Contribution

It presents the first near-constant size labeling scheme for fault-tolerant reachability in directed planar graphs with efficient query time.

Findings

Labels of size  O(1) for vertices

Query time  O(1) for reachability after vertex removal

Improves upon previous  O(n) size oracle

Abstract

We show how to assign labels of size $\tilde O(1)$ to the vertices of a directed planar graph $G$, such that from the labels of any three vertices $s,t,f$ we can deduce in $\tilde O(1)$ time whether $t$ is reachable from $s$ in the graph $G\setminus \{f\}$. Previously it was only known how to achieve $\tilde O(1)$ queries using a centralized $\tilde O(n)$ size oracle [SODA'21].