Fully dynamic arboricity maintenance

Abstract

Given an undirected graph, its arboricity is the minimum number of edge disjoint forests that its edge set can be partitioned into. We develop the first fully dynamic algorithms to determine the arboricity of a graph under edge insertions and deletions. While our insertion algorithm is based on known static algorithms to determine the arboricity, our deletion algorithm is, to the best of our knowledge, new.Our algorithms take O(mlog⁡n) time to insert or delete an edge where m is the number of edges in the graph while the best static algorithm to compute arboricity takes O(m3/2log⁡(n2/m)) time [9].We complement our upper bound with a lower bound of amortized Ω(log⁡n) time for an update for an algorithm that maintains a forest decomposition of size arboricity of the graph under edge insertions and deletions.