A Note on Dynamic Bidirected Dyck-Reachability with Cycles

Recently, Li et al. [2022] presented a dynamic Dyck-reachability algorithm for bidirected graphs. The basic idea is based on updating edge weights in a data structure called the merged graph G_m. As noted in Krishna et al. [2023], the edge deletion procedure described in the algorithm of Li et al. [2022] cannot properly update the weights in the presence of cycles in G_m. This note discusses the cycle case and the time complexity.