Abstract
Given an undirected graph, its arboricity is the minimum number of edge disjoint forests, 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 \({\tilde{O}} (m)\) time (\({\tilde{O}}\) notation ignores logarithmic factors.) 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 \(O(m^{3/2}\log (n^2/m))\) time [7].
We complement our upper bound with a lower bound result of amortized \(\varOmega (\log n)\) for any algorithm that maintains a forest decomposition of size arboricity of the graph under edge insertions and deletions.
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Notes
- 1.
Proofs of results marked \(\star \) are deferred to the full version.
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Banerjee, N., Raman, V., Saurabh, S. (2019). Fully Dynamic Arboricity Maintenance. In: Du, DZ., Duan, Z., Tian, C. (eds) Computing and Combinatorics. COCOON 2019. Lecture Notes in Computer Science(), vol 11653. Springer, Cham. https://doi.org/10.1007/978-3-030-26176-4_1
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