Optimal decremental connectivity in planar graphs

Abstract: We show an algorithm for dynamic maintenance of connectivity information in an undirected planar graph subject to edge deletions. Our algorithm may answer connectivity queries of the form `Are vertices $u$ and $v$ connected with a path?' in constant time. The queries can be intermixed with any sequence of edge deletions, and the algorithm handles all updates in $O(n)$ time. This results improves over previously known $O(n \log n)$ time algorithm.