We present a data structure for maintaining 2-edge connectivity information dynamically in an embedded planar graph. The data structure requires linear storage and preprocessing time for its construction, supports online updates (deletion of an edge or insertion of an edge consistent with the embedding) in O(log2 n) time, and answers a query (whether two vertices are in the same 2-edge-connected component) in O(log n) time. The previous best algorithm for this problem requires O(log3 n) time for updates. © 1994.
Hershberger, J., Rauch, M., & Suri, S. (1994). Data structures for two-edge connectivity in planar graphs. Theoretical Computer Science, 130(1), 139–161. https://doi.org/10.1016/0304-3975(94)90156-2